%------------------------------------------------------------------------------
% File     : ITP252_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Bounds 01246_070255
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0071_VEBT_Bounds_01246_070255 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11567 (4037 unt;1700 typ;   0 def)
%            Number of atoms       : 19865 (8166 equ)
%            Maximal formula atoms :   61 (   2 avg)
%            Number of connectives : 20072 (2110   ~; 356   |;2022   &)
%                                         (2108 <=>;13476  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   38 (   2 avg)
%            Number of FOOLs       : 1512 (1000 fml; 512 var)
%            Number of X terms     :  782 (   0  []; 624 ite; 158 let)
%            Number of types       :   14 (  13 usr)
%            Number of type conns  : 1497 (1196   >; 301   *;   0   +;   0  <<)
%            Number of predicates  :  272 ( 269 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1437 (1437 usr; 113 con; 0-8 aty)
%            Number of variables   : 33596 (30097   !; 983   ?;33596   :)
%                                         (2516  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TX1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 03:32:57.789
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
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_String_Oliteral,type,
    literal: $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_itself,type,
    itself: $tType > $tType ).

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

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

% Explicit typings (1679)
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_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_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_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_Complete__Partial__Order_Occpo,type,
    comple9053668089753744459l_ccpo: 
      !>[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_Onormalization__semidom,type,
    normal8620421768224518004emidom: 
      !>[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_Rings_Ocomm__semiring__1__cancel,type,
    comm_s4317794764714335236cancel: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
    comple6319245703460814977attice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

tff(sy_cl_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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__js____,type,
    aTP_Lamp_js: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jt____,type,
    aTP_Lamp_jt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ju____,type,
    aTP_Lamp_ju: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__kj____,type,
    aTP_Lamp_kj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__kk____,type,
    aTP_Lamp_kk: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ko____,type,
    aTP_Lamp_ko: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__kp____,type,
    aTP_Lamp_kp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kq____,type,
    aTP_Lamp_kq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

tff(sy_c_ATP_058Lamp__kt____,type,
    aTP_Lamp_kt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ku____,type,
    aTP_Lamp_ku: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kv____,type,
    aTP_Lamp_kv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kw____,type,
    aTP_Lamp_kw: 
      !>[A: $tType,B: $tType] : 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] : fun(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] : fun(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] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lk____,type,
    aTP_Lamp_lk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ll____,type,
    aTP_Lamp_ll: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__ln____,type,
    aTP_Lamp_ln: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lo____,type,
    aTP_Lamp_lo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__lq____,type,
    aTP_Lamp_lq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lr____,type,
    aTP_Lamp_lr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ls____,type,
    aTP_Lamp_ls: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lt____,type,
    aTP_Lamp_lt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lu____,type,
    aTP_Lamp_lu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lv____,type,
    aTP_Lamp_lv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lw____,type,
    aTP_Lamp_lw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

tff(sy_c_ATP_058Lamp__lz____,type,
    aTP_Lamp_lz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ma____,type,
    aTP_Lamp_ma: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mb____,type,
    aTP_Lamp_mb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__md____,type,
    aTP_Lamp_md: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__me____,type,
    aTP_Lamp_me: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

tff(sy_c_ATP_058Lamp__mh____,type,
    aTP_Lamp_mh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mi____,type,
    aTP_Lamp_mi: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mj____,type,
    aTP_Lamp_mj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mk____,type,
    aTP_Lamp_mk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

tff(sy_c_ATP_058Lamp__mq____,type,
    aTP_Lamp_mq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mr____,type,
    aTP_Lamp_mr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ms____,type,
    aTP_Lamp_ms: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

tff(sy_c_ATP_058Lamp__my____,type,
    aTP_Lamp_my: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mz____,type,
    aTP_Lamp_mz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

tff(sy_c_ATP_058Lamp__ne____,type,
    aTP_Lamp_ne: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__nw____,type,
    aTP_Lamp_nw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nx____,type,
    aTP_Lamp_nx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__nz____,type,
    aTP_Lamp_nz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oa____,type,
    aTP_Lamp_oa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ob____,type,
    aTP_Lamp_ob: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oc____,type,
    aTP_Lamp_oc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__od____,type,
    aTP_Lamp_od: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oe____,type,
    aTP_Lamp_oe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__of____,type,
    aTP_Lamp_of: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__oj____,type,
    aTP_Lamp_oj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ok____,type,
    aTP_Lamp_ok: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ol____,type,
    aTP_Lamp_ol: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__oo____,type,
    aTP_Lamp_oo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__op____,type,
    aTP_Lamp_op: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__or____,type,
    aTP_Lamp_or: 
      !>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__oy____,type,
    aTP_Lamp_oy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oz____,type,
    aTP_Lamp_oz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pa____,type,
    aTP_Lamp_pa: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pb____,type,
    aTP_Lamp_pb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pc____,type,
    aTP_Lamp_pc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__sl____,type,
    aTP_Lamp_sl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sm____,type,
    aTP_Lamp_sm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sn____,type,
    aTP_Lamp_sn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__so____,type,
    aTP_Lamp_so: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

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

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

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

tff(sy_c_ATP_058Lamp__wd____,type,
    aTP_Lamp_wd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__we____,type,
    aTP_Lamp_we: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wf____,type,
    aTP_Lamp_wf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wg____,type,
    aTP_Lamp_wg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wh____,type,
    aTP_Lamp_wh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wi____,type,
    aTP_Lamp_wi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wj____,type,
    aTP_Lamp_wj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wk____,type,
    aTP_Lamp_wk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wl____,type,
    aTP_Lamp_wl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wm____,type,
    aTP_Lamp_wm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wn____,type,
    aTP_Lamp_wn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wo____,type,
    aTP_Lamp_wo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wp____,type,
    aTP_Lamp_wp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wq____,type,
    aTP_Lamp_wq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wr____,type,
    aTP_Lamp_wr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ws____,type,
    aTP_Lamp_ws: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wt____,type,
    aTP_Lamp_wt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wu____,type,
    aTP_Lamp_wu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wv____,type,
    aTP_Lamp_wv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ww____,type,
    aTP_Lamp_ww: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wx____,type,
    aTP_Lamp_wx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wy____,type,
    aTP_Lamp_wy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wz____,type,
    aTP_Lamp_wz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xa____,type,
    aTP_Lamp_xa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xb____,type,
    aTP_Lamp_xb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xc____,type,
    aTP_Lamp_xc: 
      !>[A: $tType,B: $tType] : fun(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] : ( 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] : fun(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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

tff(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
    bNF_Ca7293521722713021262ofinal: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set(product_prod(nat,nat)) ).

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

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

tff(sy_c_BNF__Def_Orel__fun,type,
    bNF_rel_fun: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,$o)) * fun(B,fun(D,$o)) ) > fun(fun(A,B),fun(fun(C,D),$o)) ) ).

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

tff(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
    bNF_Gr4221423524335903396lImage: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(B,A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
    bNF_Gr7122648621184425601vImage: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

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

tff(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
    bNF_We413866401316099525erIncl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OordIso,type,
    bNF_Wellorder_ordIso: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_OordLess,type,
    bNF_We4044943003108391690rdLess: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Embedding_Oembed,type,
    bNF_Wellorder_embed: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

tff(sy_c_BNF__Wellorder__Embedding_OembedS,type,
    bNF_Wellorder_embedS: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

tff(sy_c_BNF__Wellorder__Embedding_Oiso,type,
    bNF_Wellorder_iso: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

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

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
    bNF_We4791949203932849705sMinim: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) * A ) > $o ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
    bNF_We1388413361240627857o_max2: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A * A ) > A ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
    bNF_We6954850376910717587_minim: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).

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

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_Basic__BNF__LFPs_Osum_Osize__sum,type,
    basic_BNF_size_sum: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * sum_sum(A,B) ) > nat ) ).

tff(sy_c_Basic__BNFs_Ofsts,type,
    basic_fsts: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(A) ) ).

tff(sy_c_Basic__BNFs_Opred__fun,type,
    basic_pred_fun: 
      !>[A: $tType,B: $tType] : ( ( fun(A,$o) * fun(B,$o) ) > fun(fun(A,B),$o) ) ).

tff(sy_c_Basic__BNFs_Osnds,type,
    basic_snds: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).

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_Oand__not__num__rel,type,
    bit_and_not_num_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

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

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_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * 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_Oand__num__rel,type,
    bit_un5425074673868309765um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

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_Oxor__num__rel,type,
    bit_un3595099601533988841um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

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_Oand__num__rel,type,
    bit_un4731106466462545111um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

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_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
    bit_un2901131394128224187um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

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_ONeg,type,
    code_Neg: num > code_integer ).

tff(sy_c_Code__Numeral_OPos,type,
    code_Pos: num > code_integer ).

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_Odup,type,
    code_dup: 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_Code__Numeral_Osub,type,
    code_sub: ( num * num ) > code_integer ).

tff(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: fun(num,int) ).

tff(sy_c_Code__Target__Int_Opositive,type,
    code_Target_positive: fun(num,int) ).

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_Complete__Partial__Order_Occpo_Oadmissible,type,
    comple1908693960933563346ssible: 
      !>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,$o)) * fun(A,$o) ) > $o ) ).

tff(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
    comple115746919287870866o_fixp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
    comple6359979572994053840erates: 
      !>[A: $tType] : ( fun(A,A) > set(A) ) ).

tff(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
    comple7512665784863727008ratesp: 
      !>[A: $tType] : ( fun(A,A) > fun(A,$o) ) ).

tff(sy_c_Complete__Partial__Order_Ochain,type,
    comple1602240252501008431_chain: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

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

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_Complex_Orcis,type,
    rcis: ( real * real ) > complex ).

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

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

tff(sy_c_Countable_Onth__item__rel,type,
    nth_item_rel: fun(nat,fun(nat,$o)) ).

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

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

tff(sy_c_Deriv_Ohas__field__derivative,type,
    has_field_derivative: 
      !>[A: $tType] : ( ( fun(A,A) * A * filter(A) ) > $o ) ).

tff(sy_c_Deriv_Ohas__vector__derivative,type,
    has_ve8173657378732805170vative: 
      !>[B: $tType] : ( ( fun(real,B) * B * filter(real) ) > $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_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
    euclid6346220572633701492n_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
    euclid7384307370059645450egment: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Extended__Nat_OeSuc,type,
    extended_eSuc: fun(extended_enat,extended_enat) ).

tff(sy_c_Extended__Nat_Oenat,type,
    extended_enat2: nat > extended_enat ).

tff(sy_c_Extended__Nat_Oenat_OAbs__enat,type,
    extended_Abs_enat: fun(option(nat),extended_enat) ).

tff(sy_c_Extended__Nat_Oenat_ORep__enat,type,
    extended_Rep_enat: fun(extended_enat,option(nat)) ).

tff(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
    extended_case_enat: 
      !>[T: $tType] : fun(fun(nat,T),fun(T,fun(extended_enat,T))) ).

tff(sy_c_Extended__Nat_Oenat_Orec__enat,type,
    extended_rec_enat: 
      !>[T: $tType] : fun(fun(nat,T),fun(T,fun(extended_enat,T))) ).

tff(sy_c_Extended__Nat_Oenat_Orec__set__enat,type,
    extend4933016492236175606t_enat: 
      !>[T: $tType] : ( ( fun(nat,T) * T * extended_enat ) > fun(T,$o) ) ).

tff(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
    extend4730790105801354508finity: 
      !>[A: $tType] : A ).

tff(sy_c_Extended__Nat_Othe__enat,type,
    extended_the_enat: extended_enat > nat ).

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Fun_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_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_Fun__Def_Opair__leq,type,
    fun_pair_leq: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

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_Obounded__quasi__semilattice__set_OF,type,
    bounde2362111253966948842tice_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A ) > fun(set(A),A) ) ).

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

tff(sy_c_GCD_Ogcd__class_Olcm,type,
    gcd_lcm: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : fun(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(fun(B,A),fun(A,fun(list(B),A))) ).

tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
    groups8242544230860333062m_list: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_Groups__List_Omonoid__list_OF,type,
    groups_monoid_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(list(A),A) ) ).

tff(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
    groups5270119922927024881d_list: 
      !>[A: $tType] : fun(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_OUniq,type,
    uniq: 
      !>[A: $tType] : ( fun(A,$o) > $o ) ).

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

tff(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
    complete_lattice_gfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
    complete_lattice_lfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
    infini527867602293511546merate: 
      !>[A: $tType] : ( ( set(A) * nat ) > A ) ).

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

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

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

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

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Int_Onat,type,
    nat2: fun(int,nat) ).

tff(sy_c_Int_Opcr__int,type,
    pcr_int: fun(product_prod(nat,nat),fun(int,$o)) ).

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

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

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__order__set,type,
    lattic4895041142388067077er_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

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

tff(sy_c_Lifting_OQuotient,type,
    quotient: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,B) * fun(B,A) * fun(A,fun(B,$o)) ) > $o ) ).

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

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

tff(sy_c_Limits_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

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

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

tff(sy_c_List_Oarg__min__list__rel,type,
    arg_min_list_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),$o)) ).

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

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

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

tff(sy_c_List_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_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,$o) * 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) * B ) > B ) ).

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

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

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

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

tff(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * 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_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * set(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
    linord329482645794927042rt_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

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

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

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

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

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

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

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

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

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

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

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

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

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_Omap__project,type,
    map_project: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > set(B) ) ).

tff(sy_c_List_Omeasures,type,
    measures: 
      !>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).

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

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_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

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

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

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

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_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_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__add,type,
    map_add: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__comp,type,
    map_comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,option(C)) * fun(A,option(B)) * A ) > option(C) ) ).

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)) * 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_ONats,type,
    semiring_1_Nats: 
      !>[A: $tType] : set(A) ).

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

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

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

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: 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,type,
    nat_prod_decode: nat > product_prod(nat,nat) ).

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

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

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),nat) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

tff(sy_c_Nat__Bijection_Osum__decode,type,
    nat_sum_decode: nat > sum_sum(nat,nat) ).

tff(sy_c_Nat__Bijection_Osum__encode,type,
    nat_sum_encode: sum_sum(nat,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] : fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))) ).

tff(sy_c_Num_Onum_Orec__num,type,
    rec_num: 
      !>[A: $tType] : fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(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_Obind,type,
    bind: 
      !>[A: $tType,B: $tType] : fun(option(A),fun(fun(A,option(B)),option(B))) ).

tff(sy_c_Option_Ocombine__options,type,
    combine_options: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * option(A) * option(A) ) > option(A) ) ).

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

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

tff(sy_c_Option_Ooption_Opred__option,type,
    pred_option: 
      !>[A: $tType] : fun(fun(A,$o),fun(option(A),$o)) ).

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

tff(sy_c_Option_Ooption_Orel__option,type,
    rel_option: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o))) ).

tff(sy_c_Option_Ooption_Oset__option,type,
    set_option: 
      !>[A: $tType] : fun(option(A),set(A)) ).

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

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

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

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

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

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

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_OAboveS,type,
    order_AboveS: 
      !>[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_Oabove,type,
    order_above: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_OaboveS,type,
    order_aboveS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * 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_Oofilter,type,
    order_ofilter: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).

tff(sy_c_Order__Relation_Opartial__order__on,type,
    order_7125193373082350890der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Orelation__of,type,
    order_relation_of: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > set(product_prod(A,A)) ) ).

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

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

tff(sy_c_Order__Relation_Owell__order__on,type,
    order_well_order_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_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

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

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

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

tff(sy_c_Orderings_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: $tType] : fun(fun(A,B),$o) ).

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

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

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

tff(sy_c_Power_Opower_Opower,type,
    power2: 
      !>[A: $tType] : ( ( A * fun(A,fun(A,A)) ) > fun(A,fun(nat,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) * product_prod(A,B) ) > product_prod(C,B) ) ).

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

tff(sy_c_Product__Type_Ointernal__case__prod,type,
    produc5280177257484947105e_prod: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * product_prod(A,B) ) > C ) ).

tff(sy_c_Product__Type_Omap__prod,type,
    product_map_prod: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(product_prod(A,B),product_prod(C,D)) ) ).

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_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_Oprod_Oswap,type,
    product_swap: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).

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

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

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

tff(sy_c_Quotient_Oquot__type_Orep,type,
    quot_rep: 
      !>[B: $tType,A: $tType] : ( fun(B,set(A)) > fun(B,A) ) ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: 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_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,$o)) ).

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: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Real_OReal,type,
    real2: fun(fun(nat,rat),real) ).

tff(sy_c_Real_Ocauchy,type,
    cauchy: fun(fun(nat,rat),$o) ).

tff(sy_c_Real_Ocr__real,type,
    cr_real: fun(fun(nat,rat),fun(real,$o)) ).

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun(fun(nat,rat),fun(real,$o)) ).

tff(sy_c_Real_Opositive,type,
    positive2: fun(real,$o) ).

tff(sy_c_Real_Oreal_OAbs__real,type,
    abs_real: fun(set(fun(nat,rat)),real) ).

tff(sy_c_Real_Oreal_ORep__real,type,
    rep_real: fun(real,set(fun(nat,rat))) ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun(fun(nat,rat),fun(fun(nat,rat),$o)) ).

tff(sy_c_Real_Orep__real,type,
    rep_real2: fun(real,fun(nat,rat)) ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun(nat,rat) > $o ).

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

tff(sy_c_Real__Vector__Spaces_Obounded__bilinear,type,
    real_V2442710119149674383linear: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > $o ) ).

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_Odependent,type,
    real_V358717886546972837endent: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Odim,type,
    real_Vector_dim: 
      !>[A: $tType] : ( set(A) > nat ) ).

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_Orepresentation,type,
    real_V7696804695334737415tation: 
      !>[A: $tType] : ( ( set(A) * A * A ) > real ) ).

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

tff(sy_c_Real__Vector__Spaces_Ospan,type,
    real_Vector_span: 
      !>[A: $tType] : ( set(A) > set(A) ) ).

tff(sy_c_Relation_ODomain,type,
    domain: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(A) ) ).

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

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_ORange,type,
    range: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(B) ) ).

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

tff(sy_c_Relation_Oconverse,type,
    converse: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(B,A)) ) ).

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

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

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_Orelcompp,type,
    relcompp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,$o)) * fun(B,fun(C,$o)) ) > fun(A,fun(C,$o)) ) ).

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

tff(sy_c_Relation_Otransp,type,
    transp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $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] : fun(A,fun(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_Onormalization__semidom__class_Onormalize,type,
    normal6383669964737779283malize: 
      !>[A: $tType] : ( 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_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : ( ( set(A) * set(A) ) > $o ) ).

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_Ois__singleton,type,
    is_singleton: 
      !>[A: $tType] : ( set(A) > $o ) ).

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_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).

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_OCode_Oabort,type,
    abort: 
      !>[A: $tType] : ( ( literal * fun(product_unit,A) ) > A ) ).

tff(sy_c_String_OLiteral,type,
    literal2: ( $o * $o * $o * $o * $o * $o * $o * literal ) > literal ).

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_Sum__Type_OInl,type,
    sum_Inl: 
      !>[A: $tType,B: $tType] : fun(A,sum_sum(A,B)) ).

tff(sy_c_Sum__Type_OInr,type,
    sum_Inr: 
      !>[B: $tType,A: $tType] : fun(B,sum_sum(A,B)) ).

tff(sy_c_Sum__Type_OPlus,type,
    sum_Plus: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).

tff(sy_c_Sum__Type_Osum_Ocase__sum,type,
    sum_case_sum: 
      !>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(B,C) * sum_sum(A,B) ) > C ) ).

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

tff(sy_c_Topological__Spaces_Otopological__space__class_Oconnected,type,
    topolo1966860045006549960nected: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent,type,
    topolo6863149650580417670ergent: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

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

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

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
    topolo6773858410816713723filter: 
      !>[A: $tType] : ( filter(A) > $o ) ).

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

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

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

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun(real,real) ).

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

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun(real,real) ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun(real,real) ).

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

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

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

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

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

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

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

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : ( 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_Opowr__real,type,
    powr_real: ( real * real ) > real ).

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] : ( A > A ) ).

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

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

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

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

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

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

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

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

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

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

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

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

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

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option(product_prod(nat,nat)) * nat * list(vEBT_VEBT) * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Orec__VEBT,type,
    vEBT_rec_VEBT: 
      !>[A: $tType] : ( ( fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))) * fun($o,fun($o,A)) * vEBT_VEBT ) > A ) ).

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 * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: ( vEBT_VEBT * extended_enat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun(product_prod(vEBT_VEBT,extended_enat),fun(product_prod(vEBT_VEBT,extended_enat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

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

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

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

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

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option(nat) * option(nat) ) > $o ).

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
    vEBT_V2048590022279873568_shift: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(option(A),fun(option(A),option(A))) ) ).

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun(option(nat),fun(option(nat),option(nat))) ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

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

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

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set(product_prod(nat,nat)) ).

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_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

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

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

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(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_Ocut,type,
    cut: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(product_prod(A,A)) * A ) > fun(A,B) ) ).

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_Zorn_OChains,type,
    chains: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).

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

tff(sy_c_Zorn_Ochains,type,
    chains2: 
      !>[A: $tType] : ( set(set(A)) > set(set(set(A))) ) ).

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

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

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

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

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

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

tff(sy_c_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_treeList____,type,
    treeList: list(vEBT_VEBT) ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (9080)
tff(fact_0__C5_Ohyps_C_I7_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),mi),ma) ).

% "5.hyps"(7)
tff(fact_1_max__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat] :
      ( vEBT_VEBT_max_in_set(Xs,Xb)
    <=> ( member(nat,Xb,Xs)
        & ! [X: nat] :
            ( member(nat,X,Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Xb) ) ) ) ).

% max_in_set_def
tff(fact_2_min__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat] :
      ( vEBT_VEBT_min_in_set(Xs,Xb)
    <=> ( member(nat,Xb,Xs)
        & ! [X: nat] :
            ( member(nat,X,Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),X) ) ) ) ).

% min_in_set_def
tff(fact_3_height__compose__summary,axiom,
    ! [Summarya: vEBT_VEBT,Info: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Summarya))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Dega,TreeLista,Summarya))) ).

% height_compose_summary
tff(fact_4__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_5__C5_OIH_C_I2_J,axiom,
    ! [Xb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c(summary,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,summary))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2))))))) ).

% "5.IH"(2)
tff(fact_6__C2_C,axiom,
    vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary),xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),vEBT_T_s_u_c_c(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),
      $ite(vEBT_vebt_succ(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))) ).

% "2"
tff(fact_7_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_8_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_9_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_10_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_11_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_12_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_a_x_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_13_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_i_n_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_14_maxt__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_a_x_t(Ta)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% maxt_bound
tff(fact_15_mint__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_t(Ta)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% mint_bound
tff(fact_16_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_17_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_18_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_19_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_20_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_21_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: 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),A3),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),A3),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_22_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_23_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_24_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_25_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_26_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))),B2) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_27_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_28_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_29_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_30_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% is_num_normalize(1)
tff(fact_31_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% divide_numeral_1
tff(fact_32_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_33_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_34_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% mult_numeral_1_right
tff(fact_35_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A3) = A3 ) ).

% mult_numeral_1
tff(fact_36_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_37_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_38_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_39_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X2: num] : Y != aa(num,num,bit0,X2)
       => ~ ! [X3: num] : Y != aa(num,num,bit1,X3) ) ) ).

% num.exhaust
tff(fact_40_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_41_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_42_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),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))),A3)),B2) ) ).

% left_add_twice
tff(fact_43_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      ( member(A,A3,aa(fun(A,$o),set(A),collect(A),P))
    <=> aa(A,$o,P,A3) ) ).

% mem_Collect_eq
tff(fact_44_Collect__mem__eq,axiom,
    ! [A: $tType,A4: set(A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_a(set(A),fun(A,$o),A4)) = A4 ).

% Collect_mem_eq
tff(fact_45_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_46_ext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X4: A] : aa(A,B,F2,X4) = aa(A,B,G,X4)
     => ( F2 = G ) ) ).

% ext
tff(fact_47_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_48_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_49_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_50_add__self__div__2,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Mb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Mb ).

% add_self_div_2
tff(fact_51_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_52_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_53_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_54_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_55_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_56_succ__list__to__short,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list(vEBT_VEBT),Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Xb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_57_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_58_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_59_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_60_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_61_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_62_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_63_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_64_option_Oinject,axiom,
    ! [A: $tType,X22: A,Y2: A] :
      ( ( aa(A,option(A),some(A),X22) = aa(A,option(A),some(A),Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_65_semiring__norm_I85_J,axiom,
    ! [Mb: num] : aa(num,num,bit0,Mb) != one2 ).

% semiring_norm(85)
tff(fact_66_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one2 != aa(num,num,bit0,Nb) ).

% semiring_norm(83)
tff(fact_67_semiring__norm_I89_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,bit1,Mb) != aa(num,num,bit0,Nb) ).

% semiring_norm(89)
tff(fact_68_semiring__norm_I88_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,bit0,Mb) != aa(num,num,bit1,Nb) ).

% semiring_norm(88)
tff(fact_69_semiring__norm_I86_J,axiom,
    ! [Mb: num] : aa(num,num,bit1,Mb) != one2 ).

% semiring_norm(86)
tff(fact_70_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one2 != aa(num,num,bit1,Nb) ).

% semiring_norm(84)
tff(fact_71_not__None__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ( Xb != none(A) )
    <=> ? [Y3: A] : Xb = aa(A,option(A),some(A),Y3) ) ).

% not_None_eq
tff(fact_72_not__Some__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ! [Y3: A] : Xb != aa(A,option(A),some(A),Y3)
    <=> ( Xb = none(A) ) ) ).

% not_Some_eq
tff(fact_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_True,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList)) ).

% True
tff(fact_88_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),L)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K),L))) ) ).

% div_mult2_numeral_eq
tff(fact_89_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Va: nat] : vEBT_T_s_u_c_c(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Va) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_90_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_T_m_a_x_t(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_91_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_T_m_i_n_t(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_92_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X22: A] : none(A) != aa(A,option(A),some(A),X22) ).

% option.distinct(1)
tff(fact_93_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X22: A] :
      ( ( Option = aa(A,option(A),some(A),X22) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_94_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X2: A] : Y != aa(A,option(A),some(A),X2) ) ).

% option.exhaust
tff(fact_95_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ? [X_1: option(A)] : aa(option(A),$o,P,X_1)
    <=> ( aa(option(A),$o,P,none(A))
        | ? [X: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X)) ) ) ).

% split_option_ex
tff(fact_96_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ! [X_1: option(A)] : aa(option(A),$o,P,X_1)
    <=> ( aa(option(A),$o,P,none(A))
        & ! [X: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X)) ) ) ).

% split_option_all
tff(fact_97_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) )
       => ( ! [A5: A,B3: B] :
              ( ( Xb = aa(A,option(A),some(A),A5) )
             => ( ( Y = aa(B,option(B),some(B),B3) )
               => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) ) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_98_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),K)) ) ).

% div_le_mono
tff(fact_99_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_100_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_101_div__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Q2) ).

% div_mult2_eq
tff(fact_102_option_Osel,axiom,
    ! [A: $tType,X22: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X22)) = X22 ).

% option.sel
tff(fact_103_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_104_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_105_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_106_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_107_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),Nb) ) ).

% numeral_Bit0_div_2
tff(fact_108_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),Nb) ) ).

% numeral_Bit1_div_2
tff(fact_109_pred__list__to__short,axiom,
    ! [Dega: nat,Xb: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_110_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_111_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_112_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_113_nat__add__left__cancel__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_114_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_115_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,V))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_116_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% div_by_1
tff(fact_117_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% bits_div_by_1
tff(fact_118_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) ) ).

% times_divide_eq_left
tff(fact_119_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_120_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ).

% divide_divide_eq_right
tff(fact_121_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_122_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ).

% times_divide_eq_right
tff(fact_123_nat__add__left__cancel__less,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_124_pred__max,axiom,
    ! [Dega: nat,Maa: nat,Xb: nat,Mia: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb)
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = aa(nat,option(nat),some(nat),Maa) ) ) ) ).

% pred_max
tff(fact_125_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,V))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_126_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_127_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))),B2) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_128_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_129__C5_Ohyps_C_I8_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),deg)) ).

% "5.hyps"(8)
tff(fact_130_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X5: A] :
        ? [Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X5) ) ).

% linordered_field_no_lb
tff(fact_131_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X5: A] :
        ? [X_12: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),X_12) ) ).

% linordered_field_no_ub
tff(fact_132_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_133_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_134_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_not_refl
tff(fact_135_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_136_less__not__refl3,axiom,
    ! [S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S),Ta)
     => ( S != Ta ) ) ).

% less_not_refl3
tff(fact_137_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A3: A] :
          ( ! [X4: A] :
              ( ! [Y5: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y5)),aa(A,B,F2,X4))
                 => aa(A,$o,P,Y5) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A3) ) ) ).

% measure_induct
tff(fact_138_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_irrefl_nat
tff(fact_139_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
             => aa(nat,$o,P,M) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% nat_less_induct
tff(fact_140_infinite__descent,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
              & ~ aa(nat,$o,P,M) ) )
     => aa(nat,$o,P,Nb) ) ).

% infinite_descent
tff(fact_141_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_142_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A3: A] :
          ( ! [X4: A] :
              ( ! [Y5: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y5)),aa(A,B,F2,X4))
                 => aa(A,$o,P,Y5) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A3) ) ) ).

% measure_induct_rule
tff(fact_143_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V2: fun(A,nat),Xb: A] :
      ( ! [X4: A] :
          ( ~ aa(A,$o,P,X4)
         => ? [Y5: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y5)),aa(A,nat,V2,X4))
              & ~ aa(A,$o,P,Y5) ) )
     => aa(A,$o,P,Xb) ) ).

% infinite_descent_measure
tff(fact_144_less__mono__imp__le__mono,axiom,
    ! [F2: 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,F2,I2)),aa(nat,nat,F2,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,F2,I)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_145_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_146_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_147_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_148_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_149_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_150_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_151_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_152_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_153_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: 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),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_less_mono1
tff(fact_154_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_155_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_156_add__less__mono,axiom,
    ! [I: nat,J: nat,K: 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),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_157_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K) ) ).

% add_lessD1
tff(fact_158_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_159_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_160_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A))) ) ) ).

% add_mono1
tff(fact_161_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).

% less_add_one
tff(fact_162_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),Mb: nat,K: nat] :
      ( ! [M2: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,M2)),aa(nat,nat,F2,N)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,Mb)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K))) ) ).

% mono_nat_linear_lb
tff(fact_163_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B2) ) ) ).

% discrete
tff(fact_164_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_165_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2) ) ) ).

% gt_half_sum
tff(fact_166_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_167_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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_168_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y4: nat] :
            ( aa(nat,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),B2) )
       => ? [X4: nat] :
            ( aa(nat,$o,P,X4)
            & ! [Y5: nat] :
                ( aa(nat,$o,P,Y5)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y5),X4) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_169_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_170_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_171_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_172_le__trans,axiom,
    ! [I: nat,J: nat,K: 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),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K) ) ) ).

% le_trans
tff(fact_173_le__refl,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Nb) ).

% le_refl
tff(fact_174_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_175_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),E)),C2) ) ).

% combine_common_factor
tff(fact_176_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% distrib_right
tff(fact_177_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% distrib_left
tff(fact_178_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_179_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_180_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_181_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_182_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),W)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_183_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W)) ) ).

% times_divide_times_eq
tff(fact_184_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% add_divide_distrib
tff(fact_185_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
    <=> ? [K2: nat] : Nb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2) ) ).

% nat_le_iff_add
tff(fact_186_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_187_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_188_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: 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),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_189_add__le__mono,axiom,
    ! [I: nat,J: nat,K: 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),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_190_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
     => ? [N: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) ) ).

% le_Suc_ex
tff(fact_191_add__leD2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% add_leD2
tff(fact_192_add__leD1,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% add_leD1
tff(fact_193_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_194_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_195_add__leE,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),Nb)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% add_leE
tff(fact_196_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: 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),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_197_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: 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),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_198_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: 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),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_199_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_200_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_201_add__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ).

% add_mult_distrib2
tff(fact_202_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K)) ).

% add_mult_distrib
tff(fact_203_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_204_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_205_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_206_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ).

% field_less_half_sum
tff(fact_207_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_208_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = Xb ) ).

% field_sum_of_halves
tff(fact_209_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% mult_1
tff(fact_210_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.right_neutral
tff(fact_211_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% add_less_cancel_right
tff(fact_212_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% add_less_cancel_left
tff(fact_213_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% add_le_cancel_right
tff(fact_214_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% add_le_cancel_left
tff(fact_215_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_216_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_217_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_218_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_219_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_220_pow__sum,axiom,
    ! [A3: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A3)) = 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_221_high__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_high(Xb,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% high_def
tff(fact_222_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_223__C5_Ohyps_C_I2_J,axiom,
    aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m) ).

% "5.hyps"(2)
tff(fact_224_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_225_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_226_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_227_bit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : vEBT_VEBT_bit_concat(H,L,D2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),D2))),L) ).

% bit_concat_def
tff(fact_228_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_229_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_230_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_231_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_232_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_233_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_234_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_235_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_236_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_237_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.assoc
tff(fact_238_mult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) ) ).

% mult.commute
tff(fact_239_mult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.left_commute
tff(fact_240_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xb: A] :
          ( ( one_one(A) = Xb )
        <=> ( Xb = one_one(A) ) ) ) ).

% one_reorient
tff(fact_241_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_242_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( K = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_243_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: A,K: A,A3: A,B2: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_244_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: A,K: A,B2: A,A3: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_245_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.assoc
tff(fact_246_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_247_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_248_add_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) ) ).

% add.commute
tff(fact_249_add_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.left_commute
tff(fact_250_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_251_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_252_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Mb: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),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_253_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_254_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_255_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_256_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_mono
tff(fact_257_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_left_mono
tff(fact_258_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ~ ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ) ).

% less_eqE
tff(fact_259_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_right_mono
tff(fact_260_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ? [C4: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) ) ) ).

% le_iff_add
tff(fact_261_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% add_le_imp_le_left
tff(fact_262_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% add_le_imp_le_right
tff(fact_263_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_264_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_265_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_266_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_strict_mono
tff(fact_267_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_strict_left_mono
tff(fact_268_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_strict_right_mono
tff(fact_269_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% add_less_imp_less_left
tff(fact_270_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% add_less_imp_less_right
tff(fact_271_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_272_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.comm_neutral
tff(fact_273_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_274_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_275_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_276_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_277_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_278_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: 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),A3),aa(num,nat,numeral_numeral(nat),Mb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb))) ) ).

% power_add_numeral
tff(fact_279_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: 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),A3),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),A3),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),A3),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_280_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_281_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_282_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_283_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: 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),A3),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),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb))) ) ).

% power_mult_numeral
tff(fact_284_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_285_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Mb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb) )
          <=> ( Mb = Nb ) ) ) ) ).

% power_inject_exp
tff(fact_286_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_287_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_288_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_289__C5_Ohyps_C_I1_J,axiom,
    vEBT_invar_vebt(summary,m) ).

% "5.hyps"(1)
tff(fact_290_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_291__C5_Ohyps_C_I4_J,axiom,
    deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) ).

% "5.hyps"(4)
tff(fact_292_geqmaxNone,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Maa),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_293_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_294_two__powr__height__bound__deg,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ).

% two_powr_height_bound_deg
tff(fact_295_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ) ).

% helpyd
tff(fact_296_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),one_one(nat)) = A3 ) ).

% power_one_right
tff(fact_297_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ) ).

% helpypredd
tff(fact_298__C5_Ohyps_C_I5_J,axiom,
    ! [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))),m))
     => ( ? [X_1: nat] : vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I3),X_1)
      <=> vEBT_V8194947554948674370ptions(summary,I3) ) ) ).

% "5.hyps"(5)
tff(fact_299_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),Nb: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M: extended_enat] :
              ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M),N)
             => aa(extended_enat,$o,P,M) )
         => aa(extended_enat,$o,P,N) )
     => aa(extended_enat,$o,P,Nb) ) ).

% enat_less_induct
tff(fact_300_L2__set__mult__ineq__lemma,axiom,
    ! [A3: real,C2: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A3),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_301_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_302_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_303_height__node,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya))) ) ).

% height_node
tff(fact_304_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_305_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: 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),A3),B2)),Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ).

% power_mult_distrib
tff(fact_306_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: 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),A3),Nb)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ).

% power_commutes
tff(fact_307_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),Nb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ).

% power_divide
tff(fact_308_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,Mb: nat,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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),A3),Mb)),Nb) ) ).

% power_mult
tff(fact_309_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ).

% one_le_power
tff(fact_310_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_311_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),Nb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ).

% power_one_over
tff(fact_312_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,Mb: nat,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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),A3),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ).

% power_add
tff(fact_313_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))) ) ) ).

% power_less_power_Suc
tff(fact_314_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))) ) ) ).

% power_gt1_lemma
tff(fact_315_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) ) ) ) ).

% power_strict_increasing
tff(fact_316_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% power_less_imp_less_exp
tff(fact_317_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) ) ) ) ).

% power_increasing
tff(fact_318_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% power_le_imp_le_exp
tff(fact_319_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% power2_eq_square
tff(fact_320_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_321_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_322_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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),A3),A3)),A3) ) ).

% power3_eq_cube
tff(fact_323_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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),A3),Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power_even_eq
tff(fact_324_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_325_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_326_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_327_self__le__ge2__pow,axiom,
    ! [K: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Mb)) ) ).

% self_le_ge2_pow
tff(fact_328_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Dega)
     => ( ( Mia != Maa )
       => ( aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Summarya)) = vEBT_VEBT_high(Maa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% summaxma
tff(fact_329_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_330_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_331__C5_OIH_C_I1_J,axiom,
    ! [X5: vEBT_VEBT] :
      ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
     => ( vEBT_invar_vebt(X5,na)
        & ! [Xa: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c(X5,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X5))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2))))))) ) ) ).

% "5.IH"(1)
tff(fact_332_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_333_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_334_power__minus__is__div,axiom,
    ! [B2: nat,A3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A3)
     => ( 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),A3),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A3)),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_335_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list(vEBT_VEBT),Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Va),Vb) = none(nat) ).

% vebt_pred.simps(4)
tff(fact_336_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_337_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))
         => ( vEBT_V8194947554948674370ptions(Ta,Xb)
           => vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_338_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))
       => vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_339_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xb) )
     => vEBT_V8194947554948674370ptions(Ta,Xb) ) ).

% maxbmo
tff(fact_340_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_V8194947554948674370ptions(Ta,Xb)
       => aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_341_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_V8194947554948674370ptions(Ta,Xb)
      <=> aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_342_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_343__C5_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X5: vEBT_VEBT] :
        ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
       => ~ ? [X_13: nat] : vEBT_V8194947554948674370ptions(X5,X_13) ) ) ).

% "5.hyps"(6)
tff(fact_344_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_345_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_346_real__divide__square__eq,axiom,
    ! [R: real,A3: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),A3)),aa(real,real,aa(real,fun(real,real),times_times(real),R),R)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),A3),R) ).

% real_divide_square_eq
tff(fact_347_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_348_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_349_height__compose__child,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list(vEBT_VEBT),Info: option(product_prod(nat,nat)),Dega: nat,Summarya: vEBT_VEBT] :
      ( member(vEBT_VEBT,Ta,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Dega,TreeLista,Summarya))) ) ).

% height_compose_child
tff(fact_350_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 )
       => ( ! [X5: vEBT_VEBT] :
              ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
             => ~ ? [X_13: nat] : vEBT_V8194947554948674370ptions(X5,X_13) )
          & ~ ? [X_13: nat] : vEBT_V8194947554948674370ptions(Summarya,X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_351_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: 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),A3),B2)),B2) = A3 ) ).

% add_diff_cancel
tff(fact_352_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: 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),A3),B2)),B2) = A3 ) ).

% diff_add_cancel
tff(fact_353_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_left
tff(fact_354_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: 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),A3),B2)),A3) = B2 ) ).

% add_diff_cancel_left'
tff(fact_355_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_right
tff(fact_356_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: 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),A3),B2)),B2) = A3 ) ).

% add_diff_cancel_right'
tff(fact_357_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_358_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_359_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_360_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).

% le_add_diff_inverse
tff(fact_361_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ) ) ).

% le_add_diff_inverse2
tff(fact_362_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A3: 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),A3),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),A3),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_363_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_364_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_365_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),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)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_366_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_367_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)
          & ? [M2: nat] :
              ( ( aa(nat,option(nat),some(nat),M2) = vEBT_vebt_mint(Summarya) )
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% mintlistlength
tff(fact_368__C5_Ohyps_C_I3_J,axiom,
    m = aa(nat,nat,suc,na) ).

% "5.hyps"(3)
tff(fact_369__C5_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [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))),m))
       => ( ( ( vEBT_VEBT_high(ma,na) = I3 )
           => vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I3),vEBT_VEBT_low(ma,na)) )
          & ! [X5: nat] :
              ( ( ( vEBT_VEBT_high(X5,na) = I3 )
                & vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I3),vEBT_VEBT_low(X5,na)) )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),mi),X5)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),ma) ) ) ) ) ) ).

% "5.hyps"(9)
tff(fact_370_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_371_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_372_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),K)),J) ).

% diff_commute
tff(fact_373_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( ( A3 = B2 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_374_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) ) ).

% diff_right_commute
tff(fact_375_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_mono
tff(fact_376_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_left_mono
tff(fact_377_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_right_mono
tff(fact_378_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_379_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_380_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_381_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_382_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_383_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% left_diff_distrib
tff(fact_384_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% right_diff_distrib
tff(fact_385_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) ) ).

% left_diff_distrib'
tff(fact_386_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% right_diff_distrib'
tff(fact_387_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A4: A,K: A,A3: A,B2: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_388_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = C2 )
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_389_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_390_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ).

% add_diff_eq
tff(fact_391_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_392_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).

% diff_add_eq
tff(fact_393_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_394_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A3 )
         => ( C2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ) ) ).

% add_implies_diff
tff(fact_395_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,C2: A,B2: A,D2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2)) ) ).

% add_diff_add
tff(fact_396_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_397_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% diff_divide_distrib
tff(fact_398_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_399_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_400_eq__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_401_le__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_402_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_403_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_404_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_405_le__diff__iff_H,axiom,
    ! [A3: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),A3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A3) ) ) ) ).

% le_diff_iff'
tff(fact_406_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_407_Nat_Odiff__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% Nat.diff_cancel
tff(fact_408_diff__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% diff_cancel2
tff(fact_409_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_410_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_411_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K)) ).

% diff_mult_distrib
tff(fact_412_diff__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ).

% diff_mult_distrib2
tff(fact_413_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_le_eq
tff(fact_414_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ) ).

% le_diff_eq
tff(fact_415_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),A3) = B2 ) ) ) ).

% diff_add
tff(fact_416_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3)) ) ) ).

% le_add_diff
tff(fact_417_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_418_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_419_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_420_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_421_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_422_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_423_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_424_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_425_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_426_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_427_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_less_eq
tff(fact_428_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ) ).

% less_diff_eq
tff(fact_429_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_430_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Xb: A,Y: A,A3: 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),A3),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),A3)),B2)) ) ).

% mult_diff_mult
tff(fact_431_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_432_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_433_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_434_less__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_435_diff__less__mono,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),A3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C2)) ) ) ).

% diff_less_mono
tff(fact_436_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: 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),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).

% less_diff_conv
tff(fact_437_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_438_le__diff__conv,axiom,
    ! [J: nat,K: 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),K)),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),K)) ) ).

% le_diff_conv
tff(fact_439_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_440_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_441_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_442_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: 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) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_443_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E)),C2)),D2) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_444_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E)),D2)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_445_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E)),C2)),D2) ) ) ).

% less_add_iff1
tff(fact_446_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E)),D2)) ) ) ).

% less_add_iff2
tff(fact_447_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_448_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),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),K)) ) ) ).

% less_diff_conv2
tff(fact_449_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_450_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_451_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_452_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_453_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_454_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_455_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_456_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_457_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_458_diff__le__diff__pow,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_459_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_460_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_461_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_462_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_463_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_464_nested__mint,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat,Va: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Nb)
     => ( ( Nb = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Mia)
         => ( ( Maa != Mia )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_465_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_12: nat] : vEBT_V8194947554948674370ptions(Summarya,X_12)
               => ( ! [X4: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_12: nat] : vEBT_V8194947554948674370ptions(X4,X_12) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_466_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q2: A,R: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q2)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q2)),R)) ) ).

% divmod_step_eq
tff(fact_467_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Dega)
       => ( vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => 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_468_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Maa)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ) ).

% member_inv
tff(fact_469__C1_C,axiom,
    vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary),xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),
      $ite(
        ( ( vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) != none(nat) )
        & vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),vEBT_T_s_u_c_c(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),
          $ite(vEBT_vebt_succ(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))) )) ).

% "1"
tff(fact_470_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)
     => ( 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)
       => ( vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_471_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_472_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))
     => ? [Info2: option(product_prod(nat,nat)),TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,Nb)),TreeList,S2) ) ).

% deg_SUcn_Node
tff(fact_473_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_474_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_475_bit__split__inv,axiom,
    ! [Xb: nat,D2: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xb,D2),vEBT_VEBT_low(Xb,D2),D2) = Xb ).

% bit_split_inv
tff(fact_476_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_477_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_478_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_479_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_480_lessI,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Nb)) ).

% lessI
tff(fact_481_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_482_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_483_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_484_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_485_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),K) ).

% Suc_diff_diff
tff(fact_486_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_487_False,axiom,
    ~ ( ( vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) != none(nat) )
      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% False
tff(fact_488_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_489_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_490_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))
       => ( vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_491_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))),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),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_492_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_493_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_494_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_495_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_496_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_497_div2__Suc__Suc,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Mb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% div2_Suc_Suc
tff(fact_498_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_499_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xb)
        & ! [Z2: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Ta),Z2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z2),Xb) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Z2),Y) ) ) ) ).

% pred_member
tff(fact_500_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Sx) ) ) ).

% succ_corr
tff(fact_501_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Px: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Px) ) ) ).

% pred_corr
tff(fact_502_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(Ta),Xb,Sx) ) ) ).

% succ_correct
tff(fact_503_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Mb)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_504_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_505_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(Ta),Xb,Sx) ) ) ).

% pred_correct
tff(fact_506_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa(nat,nat,suc,Nb) ).

% n_not_Suc_n
tff(fact_507_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xb) = aa(nat,nat,suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_508_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_509_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_510_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_511_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_512_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_513_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_514_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Nb)
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
            & aa(nat,$o,P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_515_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_516_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_517_All__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Nb)
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
           => aa(nat,$o,P,I4) ) ) ) ).

% All_less_Suc
tff(fact_518_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_519_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_520_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_521_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: 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),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_522_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,$o),P,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K3)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K3)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),K3) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_523_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_524_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_525_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R2: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( ! [X4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,X4),X4)
       => ( ! [X4: nat,Y4: nat,Z3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R2,X4),Y4)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R2,Y4),Z3)
               => aa(nat,$o,aa(nat,fun(nat,$o),R2,X4),Z3) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R2,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_526_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_527_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),N)
             => aa(nat,$o,P,M) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% full_nat_induct
tff(fact_528_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_529_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_530_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_531_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)
     => ? [M2: nat] : M4 = aa(nat,nat,suc,M2) ) ).

% Suc_le_D
tff(fact_532_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_533_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_534_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_535_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_536_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_537_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A3: nat] :
      ( ( A4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A3) )
     => ( aa(nat,nat,suc,A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A3)) ) ) ).

% nat_arith.suc1
tff(fact_538_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K: nat,I: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I)) ) ) ).

% zero_induct_lemma
tff(fact_539_Suc__mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_540_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_541_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,N3)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_542_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,Mb: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Mb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_543_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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),A3),Nb)),A3) ) ).

% power_Suc2
tff(fact_544_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ).

% power_Suc
tff(fact_545_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,N))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N3)),aa(nat,A,F2,Nb)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_546_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,N3)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_547_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_548_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_549_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_550_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_551_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_552_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_553_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_554_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_555_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_556_less__imp__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_imp_Suc_add
tff(fact_557_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
    <=> ? [K2: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2)) ) ).

% less_iff_Suc_add
tff(fact_558_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_559_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_560_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ~ ! [Q3: nat] : Nb != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q3)) ) ).

% less_natE
tff(fact_561_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_562_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_563_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_564_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_565_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_566_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_567_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_568_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_569_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_570_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_571_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_572_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))
         => ( ! [M2: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),one2)
           => ( ! [M2: 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,M2)),aa(num,num,bit0,N))
             => ( ! [M2: 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,M2)),aa(num,num,bit1,N))
               => ( ! [M2: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),one2)
                 => ( ! [M2: 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,M2)),aa(num,num,bit0,N))
                   => ~ ! [M2: 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,M2)),aa(num,num,bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_573_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,Nb))) ) ) ).

% power_gt1
tff(fact_574_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_575_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_576_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q2)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_577_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_578_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb)))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_579_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% power_odd_eq
tff(fact_580_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_1: nat] : vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2),X_1)
                    <=> 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_12: nat] : vEBT_V8194947554948674370ptions(X4,X_12) ) )
                 => ( 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 )
                                 => vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( vEBT_VEBT_high(X4,Nb) = I2 )
                                      & 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_581_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_12: nat] : vEBT_V8194947554948674370ptions(Summarya,X_12)
               => ( ! [X4: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_12: nat] : vEBT_V8194947554948674370ptions(X4,X_12) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_582_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_1: nat] : vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2),X_1)
                    <=> 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_12: nat] : vEBT_V8194947554948674370ptions(X4,X_12) ) )
                 => ( 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 )
                                 => vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( vEBT_VEBT_high(X4,Nb) = I2 )
                                      & 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_583_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list(vEBT_VEBT),Xb: nat] :
      ( vEBT_V5917875025757280293ildren(Nb,TreeLista,Xb)
    <=> 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_584_real__average__minus__first,axiom,
    ! [A3: real,B2: real] : aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),B2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),A3) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% real_average_minus_first
tff(fact_585_real__average__minus__second,axiom,
    ! [B2: real,A3: real] : aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),A3)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),A3) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% real_average_minus_second
tff(fact_586_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_587__C0_C,axiom,
    vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary),xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),
      $let(
        maxlow: option(nat),
        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
          $ite(
            ( ( maxlow != none(nat) )
            & vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),maxlow) ),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
            $let(
              sc: option(nat),
              sc:= vEBT_vebt_succ(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(nat))),
                $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),aa(option(nat),nat,the2(nat),sc)))))) ) )) )) ).

% "0"
tff(fact_588_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_589_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_590_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_591_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_592_pred__lesseq__max,axiom,
    ! [Dega: nat,Xb: nat,Maa: nat,Mia: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa)
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb),aa(nat,option(nat),some(nat),Mia),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_593_pred__less__length__list,axiom,
    ! [Dega: nat,Xb: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb),aa(nat,option(nat),some(nat),Mia),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_594_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_595_succ__greatereq__min,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_596_succ__less__length__list,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list(vEBT_VEBT),Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Xb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_597_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_aa(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_598_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_599_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_600_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_601_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_602_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_603_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,Xb,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xb)
        & ! [X: nat] :
            ( member(nat,X,Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Xb)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_604_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)
        & ! [X: nat] :
            ( member(nat,X,Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),X)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_605_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,$o)),Uv2: option(A)] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,$o)),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))),Uw2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F3: fun(A,fun(A,$o)),X4: A,Y4: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),F3),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),X4)),aa(A,option(A),some(A),Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_606_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,A)),Uv2: option(A)] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,A)),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))),Uw2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F3: fun(A,fun(A,A)),A5: A,B3: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),F3),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),A5)),aa(A,option(A),some(A),B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_607_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,F2),aa(A,option(A),some(A),A3)),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F2,A3),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_608_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uv: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uu),none(A)),Uv) = none(A) ).

% VEBT_internal.option_shift.simps(1)
tff(fact_609_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
        aa(nat,option(nat),some(nat),Mia),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Summarya,h),
                    $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_610_vebt__pred_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
        aa(nat,option(nat),some(nat),Maa),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                  $let(
                    pr: option(nat),
                    pr:= vEBT_vebt_pred(Summarya,h),
                    $ite(
                      pr = none(nat),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb),aa(nat,option(nat),some(nat),Mia),none(nat)),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_611_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)))),
                    $let(
                      maxlow: option(nat),
                      maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                        $ite(
                          ( ( maxlow != none(nat) )
                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                          $let(
                            sc: option(nat),
                            sc:= vEBT_vebt_succ(Summarya,h),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summarya,h))),one_one(nat))),
                              $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_612_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw: fun(A,fun(A,A)),V: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uw),aa(A,option(A),some(A),V)),none(A)) = none(A) ).

% VEBT_internal.option_shift.simps(2)
tff(fact_613_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) ) ) )
         => ~ ! [A5: A] :
                ( ( Xaa = aa(A,option(A),some(A),A5) )
               => ! [B3: A] :
                    ( ( Xba = aa(A,option(A),some(A),B3) )
                   => ( Y != aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Xb,A5),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_614_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = none(nat) ).

% vebt_maxt.simps(2)
tff(fact_615_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = none(nat) ).

% vebt_mint.simps(2)
tff(fact_616_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
              $false,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_617_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summarya,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_618_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
        $ite(
          Xb = Mia,
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
            $ite(
              Xb = Maa,
              one_one(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
                  one_one(nat),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                    $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
                      one_one(nat),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))),
                        $let(
                          h: nat,
                          h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),one_one(nat)) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_619_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(vEBT_VEBT,fun(nat,fun(nat,$o)),Ta),Xb)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_620_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ad(vEBT_VEBT,fun(nat,fun(nat,$o)),Ta),Xb)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_621_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,V),TreeLista,Vc),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_622_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeLista: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeLista,S),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_623_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list(vEBT_VEBT),Vd: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeLista,Vd),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_624_member__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% member_bound_height
tff(fact_625_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_626_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( vEBT_V8194947554948674370ptions(Ta,Xb)
    <=> ( vEBT_V5719532721284313246member(Ta,Xb)
        | vEBT_VEBT_membermima(Ta,Xb) ) ) ).

% both_member_options_def
tff(fact_627_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_628_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_629_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Va: nat] : vEBT_T_s_u_c_c2(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Va) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_630_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xb) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_631_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_632_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_633_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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_634_insert__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2))))))) ) ).

% insert_bound_height
tff(fact_635_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,B3: $o] : Xb = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [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 != one_one(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 != one_one(nat) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                     => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                                        $let(
                                          maxlow: option(nat),
                                          maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                            $ite(
                                              ( ( maxlow != none(nat) )
                                              & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                              $let(
                                                sc: option(nat),
                                                sc:= vEBT_vebt_succ(Summary,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary,h))),one_one(nat))),
                                                  $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                      one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_636_all__set__conv__all__nth,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,$o,P,X) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_637_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_638_in__set__conv__nth,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_639_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_640_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_641_vebt__pred_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A5: $o] :
              ( ? [Uw2: $o] : Xb = vEBT_Leaf((A5),(Uw2))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
         => ( ! [A5: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A5),(B3)) )
               => ( ? [Va3: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va3))
                 => ( Y != $ite(
                        (B3),
                        aa(nat,option(nat),some(nat),one_one(nat)),
                        $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va2)
               => ( Y != none(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve)
                 => ( Y != none(nat) ) )
               => ( ( ? [V3: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                              aa(nat,option(nat),some(nat),Ma),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                        aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                        $let(
                                          pr: option(nat),
                                          pr:= vEBT_vebt_pred(Summary,h),
                                          $ite(
                                            pr = none(nat),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xaa),aa(nat,option(nat),some(nat),Mi),none(nat)),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                    none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_642_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_643_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_0_not
tff(fact_644_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_645_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_leaf
tff(fact_646_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_647_Leaf__0__not,axiom,
    ! [A3: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A3),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_648_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
    <=> ? [A6: $o,B5: $o] : Ta = vEBT_Leaf((A6),(B5)) ) ).

% deg1Leaf
tff(fact_649_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
     => ? [A5: $o,B3: $o] : Ta = vEBT_Leaf((A5),(B3)) ) ).

% deg_1_Leaf
tff(fact_650_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( Nb = one_one(nat) )
       => ? [A5: $o,B3: $o] : Ta = vEBT_Leaf((A5),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_651_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_652_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_653_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_654_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_655_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_656_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_657_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_658_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_659_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.right_neutral
tff(fact_660_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_661_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = A3 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_662_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = A3 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_663_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_664_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_665_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_666_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_667_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add_0
tff(fact_668_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).

% diff_self
tff(fact_669_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).

% diff_0_right
tff(fact_670_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_diff
tff(fact_671_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).

% diff_zero
tff(fact_672_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_673_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_674_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_675_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% bits_div_0
tff(fact_676_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_677_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_678_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_679_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_680_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% div_0
tff(fact_681_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_682_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_683_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A3) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_684_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),A3) ).

% bot_nat_0.extremum
tff(fact_685_le0,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% le0
tff(fact_686_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_687_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_688_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_689_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_690_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_691_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_692_mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_693_mult__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_694_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_695_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_696_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_697_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_698_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_699_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_700_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_701_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_702_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel1
tff(fact_703_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel2
tff(fact_704_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_705_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_706_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% diff_ge_0_iff_ge
tff(fact_707_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).

% diff_gt_0_iff_gt
tff(fact_708_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_709_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_710_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_711_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_712_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_713_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_714_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% div_mult_mult1
tff(fact_715_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% div_mult_mult2
tff(fact_716_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ).

% div_mult_mult1_if
tff(fact_717_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_718_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_719_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),A3) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_720_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_721_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_722_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),B2) = A3 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_723_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_724_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_725_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_726_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% div_self
tff(fact_727_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_728_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% divide_self
tff(fact_729_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_730_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = one_one(A) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_731_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_732_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_733_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_734_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_735_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_736_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% power_Suc0_right
tff(fact_737_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_738_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_739_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_740_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_741_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_742_less__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),one_one(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_one
tff(fact_743_div__by__Suc__0,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,suc,zero_zero(nat))) = Mb ).

% div_by_Suc_0
tff(fact_744_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_745_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_746_mult__less__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_747_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_748_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = zero_zero(nat) ) ) ).

% div_less
tff(fact_749_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_750_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_751_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_752_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_753_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_754_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = $ite(K = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_755_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_756_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% zero_le_divide_1_iff
tff(fact_757_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% zero_less_divide_1_iff
tff(fact_758_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_759_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_760_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_761_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_762_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_763_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) = A3 )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_764_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)) = B2,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_765_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_766_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_767_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self1
tff(fact_768_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self2
tff(fact_769_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self3
tff(fact_770_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self4
tff(fact_771_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% power_eq_0_iff
tff(fact_772_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_773_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_774_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_775_mult__le__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_776_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Mb)),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_777_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_778_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_779_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_780_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_781_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_782_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_783_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_784_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_785_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_786_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_787_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_788_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_789_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_790_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
            <=> ( Xb = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_791_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_792_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_793_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_794_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: A] :
          ( ( zero_zero(A) = Xb )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_795_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o,Xb: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A3),(B2)),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(Xb = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_796_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B3: $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((A5),(B3))),X4)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Ux2)
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S2: 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,S2)),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_797_invar__vebt_Ointros_I1_J,axiom,
    ! [A3: $o,B2: $o] : vEBT_invar_vebt(vEBT_Leaf((A3),(B2)),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_798_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_799_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_800_vebt__member_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A3),(B2))),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A3),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_801_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_802_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_803_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),D3)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList,Summary)),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_804_vebt__insert_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A3),(B2)),Xb) = $ite(
        Xb = zero_zero(nat),
        vEBT_Leaf($true,(B2)),
        $ite(Xb = one_one(nat),vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_805_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A3),(B2)),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A3),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_806_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv)),zero_zero(nat)) = none(nat) ).

% vebt_pred.simps(1)
tff(fact_807_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uu),(B2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_808_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_809_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_810_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_811_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_812_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_813_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_814_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_815_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_816_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_817_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) )
         => ( ( A3 != zero_zero(A) )
            & ( B2 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_818_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
         => ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_819_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_820_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
          <=> ( A3 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_821_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
          <=> ( A3 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_822_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_823_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% comm_monoid_add_class.add_0
tff(fact_824_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.comm_neutral
tff(fact_825_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add.group_left_neutral
tff(fact_826_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_827_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,Nb: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_828_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_829_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_830_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_831_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_832_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_833_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_834_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_835_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Mb: nat,Nb: nat] :
      ( ! [X4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X4),zero_zero(nat))
     => ( ! [Y4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y4))
       => ( ! [X4: nat,Y4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X4),Y4)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X4)),aa(nat,nat,suc,Y4)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_836_zero__induct,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_837_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa(nat,nat,suc,Mb) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_838_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_neq_Suc
tff(fact_839_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_not_Suc
tff(fact_840_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ? [M2: nat] : Nb = aa(nat,nat,suc,M2) ) ).

% not0_implies_Suc
tff(fact_841_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)
             => ? [Y5: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y5)),aa(A,nat,V2,X4))
                  & ~ aa(A,$o,P,Y5) ) ) )
       => aa(A,$o,P,Xb) ) ) ).

% infinite_descent0_measure
tff(fact_842_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,$o,P,N)
             => ? [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
                  & ~ aa(nat,$o,P,M) ) ) )
       => aa(nat,$o,P,Nb) ) ) ).

% infinite_descent0
tff(fact_843_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_844_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_zeroE
tff(fact_845_not__less0,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% not_less0
tff(fact_846_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_847_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_848_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_849_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero(nat),Ts,S),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_850_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_851_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),zero_zero(nat))
    <=> ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_852_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),zero_zero(nat))
     => ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_853_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_854_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_855_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_856_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_857_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_858_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
    <=> ( ( K = zero_zero(nat) )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_859_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_860_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$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)
               => ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_861_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,A3: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$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),A3),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
              <=> ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_862_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : vEBT_T_s_u_c_c(vEBT_Leaf((Uu),(B2)),zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_863_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_864_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o] :
      vEBT_T_m_i_n_t(vEBT_Leaf((A3),(B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_865_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_866_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(nat,$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),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ) ).

% power_strict_mono
tff(fact_867_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu),(B2)),zero_zero(nat)) = $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ).

% vebt_succ.simps(1)
tff(fact_868_vebt__maxt_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A3),(B2))) = $ite(
        (B2),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A3),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_maxt.simps(1)
tff(fact_869_vebt__pred_Osimps_I2_J,axiom,
    ! [A3: $o,Uw: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A3),(Uw)),aa(nat,nat,suc,zero_zero(nat))) = $ite((A3),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ).

% vebt_pred.simps(2)
tff(fact_870_vebt__mint_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A3),(B2))) = $ite(
        (A3),
        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_871_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_872_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_873_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_874_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_875_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)) ) ).

% zero_le_square
tff(fact_876_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$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),A3),B2)) ) ) ).

% split_mult_pos_le
tff(fact_877_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_878_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$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),A3),B2)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_879_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono
tff(fact_880_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_881_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono
tff(fact_882_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_883_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$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),A3),B2)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_884_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3),B2)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_885_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3),B2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_886_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$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),A3),B2)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_887_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_888_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_889_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_890_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_891_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_892_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_893_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_894_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_895_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ) ) ).

% add_decreasing
tff(fact_896_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_increasing
tff(fact_897_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ) ) ).

% add_decreasing2
tff(fact_898_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_increasing2
tff(fact_899_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3),B2)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_900_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$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),A3),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_901_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_902_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_903_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$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),A3),B2)) ) ) ) ).

% mult_neg_neg
tff(fact_904_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),zero_zero(A)) ) ).

% not_square_less_zero
tff(fact_905_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_less_0_iff
tff(fact_906_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$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),A3),B2)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_907_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3),B2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_908_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3),B2)) ) ) ) ).

% mult_pos_pos
tff(fact_909_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_910_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_911_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos
tff(fact_912_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos2
tff(fact_913_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_914_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_915_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_916_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono
tff(fact_917_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_918_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_919_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_920_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_921_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_922_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_923_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_924_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_925_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$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),A3),B2)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_926_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3),B2)) ) ) ) ).

% add_pos_pos
tff(fact_927_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ ! [C3: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
               => ( C3 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_928_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% pos_add_strict
tff(fact_929_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_930_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_931_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_le_0_iff
tff(fact_932_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_right_mono
tff(fact_933_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_934_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_935_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_936_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_937_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_938_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_939_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_940_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_941_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_942_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_943_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_944_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_less_0_iff
tff(fact_945_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_pos_pos
tff(fact_946_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_947_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_948_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_neg_neg
tff(fact_949_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ).

% zero_le_power
tff(fact_950_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono
tff(fact_951_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ).

% zero_less_power
tff(fact_952_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_953_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_954_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_955_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 ) ) ) ) ).

% divide_eq_imp
tff(fact_956_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 )
           => ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_957_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_958_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_959_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = one_one(A) )
          <=> ( A3 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_960_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_961_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_962_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_963_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ? [M2: nat] : Nb = aa(nat,nat,suc,M2) ) ).

% gr0_implies_Suc
tff(fact_964_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_965_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_966_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
            & aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_967_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_968_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_969_option_Osize_I4_J,axiom,
    ! [A: $tType,X22: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_970_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),Nb)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K3)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,K3) ) ) ) ).

% ex_least_nat_le
tff(fact_971_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_972_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ? [K3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_973_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_974_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_975_nat__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_976_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_977_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: 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)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ) ).

% mult_less_mono1
tff(fact_978_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: 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)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ) ).

% mult_less_mono2
tff(fact_979_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_980_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_981_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_982_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_983_vebt__pred_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))
     => ( ! [A5: $o,Uw2: $o] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A5: $o,B3: $o,Va3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va3)))
         => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va2: vEBT_VEBT,Vb2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va2)),Vb2)
           => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve)),Vf)
             => ( ! [V3: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Vj)
               => ~ ! [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_pred.cases
tff(fact_984_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,B3: $o] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B3))),zero_zero(nat))
     => ( ! [Uv2: $o,Uw2: $o,N: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))
       => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Va2)
         => ( ! [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) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_985_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B3: $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((A5),(B3))),X4)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: 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),Ts2,S2)),X4)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: 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)),Ts2,S2)),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) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_986_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B3: $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((A5),(B3))),X4)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: 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)),Uu2,Uv2,Uw2)),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) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_987_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Uw2)
     => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT,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_988_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts,S),Xb) = vEBT_Node(Info,zero_zero(nat),Ts,S) ).

% vebt_insert.simps(2)
tff(fact_989_vebt__pred_Osimps_I3_J,axiom,
    ! [A3: $o,B2: $o,Va: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A3),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        (B2),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A3),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_pred.simps(3)
tff(fact_990_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Ux: nat] : ~ vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero(nat),Uv,Uw),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_991_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_992_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o,Xb: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A3),(B2)),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
        $ite(Xb = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_993_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_T_s_u_c_c(vEBT_Leaf((Uv),(Uw)),aa(nat,nat,suc,Nb)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_994_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv),(Uw)),aa(nat,nat,suc,Nb)) = none(nat) ).

% vebt_succ.simps(2)
tff(fact_995_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_996_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_997_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_998_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_999_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1000_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1001_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1002_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1003_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1004_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1005_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1006_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1007_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1008_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1009_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$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),A3),B2)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_1010_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3),B2)) ) ) ) ).

% add_nonneg_pos
tff(fact_1011_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$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),A3),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_1012_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3),B2)) ) ) ) ).

% add_pos_nonneg
tff(fact_1013_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_1014_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_1015_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_1016_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_1017_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_1018_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
         => ( aa(A,$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),A3),B2)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_1019_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),A3) ) ) ) ).

% mult_left_le
tff(fact_1020_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_positive
tff(fact_1021_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1022_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1023_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1024_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1025_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1026_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% divide_le_cancel
tff(fact_1027_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less2
tff(fact_1028_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less
tff(fact_1029_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_le
tff(fact_1030_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_1031_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_1032_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uv),(Uw)),aa(nat,nat,suc,Nb)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_1033_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_1034_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_1035_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
         => ( aa(A,$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),A3),B2) ) ) ) ).

% power_less_imp_less_base
tff(fact_1036_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_1037_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1038_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1039_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1040_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1041_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1042_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_1043_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1044_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1045_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_1046_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_1047_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% divide_less_eq
tff(fact_1048_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1049_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1050_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_1051_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2),aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1052_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B2,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1053_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z)),B2) = $ite(Z = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1054_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1055_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1056_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_1057_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_1058_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1059_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1060_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)))
         => ( aa(A,$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),A3),B2) ) ) ) ).

% power_le_imp_le_base
tff(fact_1061_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( A3 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1062_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1063_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1064_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1065_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),Y)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1066_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1067_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1068_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o] :
      vEBT_T_m_a_x_t(vEBT_Leaf((A3),(B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((B2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_1069_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_1070_num_Osize_I5_J,axiom,
    ! [X22: num] : aa(num,nat,size_size(num),aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_1071_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_1072_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_1073_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Nb)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),K3)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,aa(nat,nat,suc,K3)) ) ) ) ).

% ex_least_nat_less
tff(fact_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_nat__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_1081_nat__diff__split,axiom,
    ! [P: fun(nat,$o),A3: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D4: nat] :
            ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
           => aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split
tff(fact_1082_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),A3: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D4: nat] :
              ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
              & ~ aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_1083_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_1084_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Mb)) ) ) ).

% div_le_mono2
tff(fact_1085_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),K)) ) ).

% power_gt_expt
tff(fact_1086_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_1087_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = Mb )
      <=> ( Nb = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1088_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_1089_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Q2)),Nb)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_1090_nat__mult__div__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_1091_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xb) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_1092_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_1093_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xb) = Y )
     => ( ! [A5: $o] :
            ( ? [B3: $o] : Xb = vEBT_Leaf((A5),(B3))
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((A5),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_1094_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_1095_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A5: $o,B3: $o] : Xb != vEBT_Leaf((A5),(B3))
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_1096_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1097_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1098_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1099_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1100_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1101_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1102_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1103_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1104_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_1105_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% divide_le_eq
tff(fact_1106_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1107_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_1108_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_1109_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1110_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1111_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_1112_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1113_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1114_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1115_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1116_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1117_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xb: A,A3: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A3)
           => ( aa(A,$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))),A3) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1118_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => ( Y != $ite(
                  (B3),
                  aa(nat,option(nat),some(nat),one_one(nat)),
                  $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_1119_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1120_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1121_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_le_eq
tff(fact_1122_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => ( Y != $ite(
                  (A5),
                  aa(nat,option(nat),some(nat),zero_zero(nat)),
                  $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_1123_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ) ).

% power_Suc_less
tff(fact_1124_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_less_eq
tff(fact_1125_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,Nb))),A3) ) ) ) ).

% power_Suc_le_self
tff(fact_1126_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,Nb))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_1127_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1128_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ) ) ).

% power_decreasing
tff(fact_1129_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_1130_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
         => ( aa(nat,$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),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ) ).

% self_le_power
tff(fact_1131_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(nat,$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),A3),Nb)) ) ) ) ).

% one_less_power
tff(fact_1132_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_1133_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_1134_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,Nb: nat,Mb: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ) ) ).

% power_diff
tff(fact_1135_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_1136_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_1137_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_1138_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_1139_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_1140_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_1141_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Q2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1142_split__div,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I4)),J3) )
               => aa(nat,$o,P,I4) ) ) ) ) ) ).

% split_div
tff(fact_1143_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_1144_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_1145_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_1146_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_1147_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve2: nat] : vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc,Vd),Ve2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_1148_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_1149_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_1150_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve2),Vf2) = none(nat) ).

% vebt_pred.simps(5)
tff(fact_1151_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve2: nat] : vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc,Vd),Ve2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_1152_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xb: A,A3: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A3)
           => ( aa(A,$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))),A3) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1153_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1154_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1155_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ).

% half_gt_zero
tff(fact_1156_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% half_gt_zero_iff
tff(fact_1157_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R: A,S: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R),S)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U))),S))),V) ) ) ) ) ).

% scaling_mono
tff(fact_1158_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% zero_le_power2
tff(fact_1159_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_1160_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_1161_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_1162_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_1163_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_1164_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_1165_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,Mb: nat,Nb: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1166_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_1167_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_1168_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_1169_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_1170_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Nb: nat,A3: A] :
          ( aa(nat,$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),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb) ) ) ) ).

% power_minus_mult
tff(fact_1171_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_1172_split__div_H,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)),Mb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q4)))
            & aa(nat,$o,P,Q4) ) ) ) ).

% split_div'
tff(fact_1173_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2),Vi2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_1174_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_1175_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2),Vj2) = none(nat) ).

% vebt_pred.simps(6)
tff(fact_1176_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2),Vi2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_1177_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_1178_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_1179_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_1180_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_1181_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_1182_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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_1183_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_1184_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% div_2_gt_zero
tff(fact_1185_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1186_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_1187_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S2)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_1188_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A5),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S2)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_1189_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A5),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S2)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_1190_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz),Xb) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_1191_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xb) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((B3),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_1192_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1193_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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_1194_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_1195_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_1196_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A5),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                        $false,
                        $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_1197_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va2,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_1198_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va2,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_1199_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),Xb) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_1200_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A5),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_1201_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_1202_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeLista,Summarya),Xb) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_1203_insersimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ~ ? [X_12: nat] : vEBT_V8194947554948674370ptions(Ta,X_12)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insersimp
tff(fact_1204_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_1205_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xb,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xb = vEBT_Leaf((A5),(B3))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) )
         => ( ( ? [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 != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) )
           => ( ( ? [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 != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                          $ite(
                            Xaa = Mi,
                            one_one(nat),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Ma,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),one_one(nat)) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_1206_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A5: $o,B3: $o] : A1 = vEBT_Leaf((A5),(B3))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X5: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => vEBT_invar_vebt(X5,N) )
                 => ( vEBT_invar_vebt(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) )
                     => ( ( M2 = N )
                       => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                         => ( ~ ? [X_13: nat] : vEBT_V8194947554948674370ptions(Summary,X_13)
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X_13: nat] : vEBT_V8194947554948674370ptions(X5,X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X5: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => vEBT_invar_vebt(X5,N) )
                   => ( vEBT_invar_vebt(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) )
                       => ( ( M2 = aa(nat,nat,suc,N) )
                         => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                           => ( ~ ? [X_13: nat] : vEBT_V8194947554948674370ptions(Summary,X_13)
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X_13: nat] : vEBT_V8194947554948674370ptions(X5,X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: 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 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_invar_vebt(X5,N) )
                     => ( vEBT_invar_vebt(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) )
                         => ( ( M2 = N )
                           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                             => ( ! [I3: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2))
                                   => ( ? [X_1: nat] : vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3),X_1)
                                    <=> vEBT_V8194947554948674370ptions(Summary,I3) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X5: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                       => ~ ? [X_13: nat] : vEBT_V8194947554948674370ptions(X5,X_13) ) )
                                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I3: nat] :
                                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2))
                                             => ( ( ( vEBT_VEBT_high(Ma,N) = I3 )
                                                 => vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3),vEBT_VEBT_low(Ma,N)) )
                                                & ! [X5: nat] :
                                                    ( ( ( vEBT_VEBT_high(X5,N) = I3 )
                                                      & vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3),vEBT_VEBT_low(X5,N)) )
                                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X5)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X5: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_invar_vebt(X5,N) )
                       => ( vEBT_invar_vebt(Summary,M2)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2) )
                           => ( ( M2 = aa(nat,nat,suc,N) )
                             => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                               => ( ! [I3: nat] :
                                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2))
                                     => ( ? [X_1: nat] : vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3),X_1)
                                      <=> vEBT_V8194947554948674370ptions(Summary,I3) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X5: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                         => ~ ? [X_13: nat] : vEBT_V8194947554948674370ptions(X5,X_13) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I3: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2))
                                               => ( ( ( vEBT_VEBT_high(Ma,N) = I3 )
                                                   => vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3),vEBT_VEBT_low(Ma,N)) )
                                                  & ! [X5: nat] :
                                                      ( ( ( vEBT_VEBT_high(X5,N) = I3 )
                                                        & vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3),vEBT_VEBT_low(X5,N)) )
                                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X5)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_1207_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A6: $o,B5: $o] : A1 = vEBT_Leaf((A6),(B5))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N4) )
            & vEBT_invar_vebt(Summary2,N4)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ~ ? [X_1: nat] : vEBT_V8194947554948674370ptions(Summary2,X_1)
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_1: nat] : vEBT_V8194947554948674370ptions(X,X_1) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N4) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N4))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(nat,nat,suc,N4)) )
            & ~ ? [X_1: nat] : vEBT_V8194947554948674370ptions(Summary2,X_1)
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_1: nat] : vEBT_V8194947554948674370ptions(X,X_1) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N4) )
            & vEBT_invar_vebt(Summary2,N4)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4))
               => ( ? [X_1: nat] : vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4),X_1)
                <=> vEBT_V8194947554948674370ptions(Summary2,I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_1: nat] : vEBT_V8194947554948674370ptions(X,X_1) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4))
                 => ( ( ( vEBT_VEBT_high(Ma2,N4) = I4 )
                     => vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4),vEBT_VEBT_low(Ma2,N4)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N4) = I4 )
                          & vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4),vEBT_VEBT_low(X,N4)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N4) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N4))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(nat,nat,suc,N4)) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)))
               => ( ? [X_1: nat] : vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4),X_1)
                <=> vEBT_V8194947554948674370ptions(Summary2,I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_1: nat] : vEBT_V8194947554948674370ptions(X,X_1) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)))
                 => ( ( ( vEBT_VEBT_high(Ma2,N4) = I4 )
                     => vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4),vEBT_VEBT_low(Ma2,N4)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N4) = I4 )
                          & vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4),vEBT_VEBT_low(X,N4)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1208_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_1209_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
         => ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P,I4),X_1) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I4),aa(nat,A,nth(A,Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_1210_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1211_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,B3: $o] : Xb = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [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 != one_one(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 != one_one(nat) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            one_one(nat),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
                                  one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_1212_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                      $let(
                                        sc: option(nat),
                                        sc:= vEBT_vebt_succ(Summary,h),
                                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                  none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_1213_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xb,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xb = vEBT_Leaf((A5),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [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 != one_one(nat) ) )
           => ( ( ? [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 != one_one(nat) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                          $ite(
                            Xaa = Mi,
                            zero_zero(nat),
                            $ite(
                              Xaa = Ma,
                              zero_zero(nat),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                zero_zero(nat),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                  zero_zero(nat),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xaa)
                                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma) ),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(nat)) ),
                                    zero_zero(nat) ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_1214_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          Xb = Mia,
          zero_zero(nat),
          $ite(
            Xb = Maa,
            zero_zero(nat),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
              zero_zero(nat),
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
                zero_zero(nat),
                $ite(
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb)
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Maa) ),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(nat)) ),
                  zero_zero(nat) ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_1215_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_1216_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% set_bit_0
tff(fact_1217_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [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 = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),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 = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                       => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),
                                        $ite(
                                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                                            $let(
                                              maxlow: option(nat),
                                              maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                                $ite(
                                                  ( ( maxlow != none(nat) )
                                                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                                  $let(
                                                    sc: option(nat),
                                                    sc:= vEBT_vebt_succ(Summary,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary,h))),one_one(nat))),
                                                      $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                          one_one(nat) )) ) ) )) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_1218_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                          $let(
                                            sc: option(nat),
                                            sc:= vEBT_vebt_succ(Summary,h),
                                            $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                      none(nat) ) ) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_1219_vebt__pred_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A5),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A5),(B3)) )
                 => ! [Va3: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va3)) )
                     => ( ( Y = $ite(
                              (B3),
                              aa(nat,option(nat),some(nat),one_one(nat)),
                              $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va3)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va2) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va2)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve)),Xaa)) ) )
                 => ( ! [V3: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = none(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                  aa(nat,option(nat),some(nat),Ma),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                            $let(
                                              pr: option(nat),
                                              pr:= vEBT_vebt_pred(Summary,h),
                                              $ite(
                                                pr = none(nat),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xaa),aa(nat,option(nat),some(nat),Mi),none(nat)),
                                                aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                        none(nat) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_1220_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_1221_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [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 = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),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 = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
                                      one_one(nat) ) ) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_1222_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Ta)
     => ? [X_12: nat] : vEBT_V8194947554948674370ptions(Ta,X_12) ) ).

% not_min_Null_member
tff(fact_1223_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_1224_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Ta)
     => ( vEBT_vebt_mint(Ta) = none(nat) ) ) ).

% minNullmin
tff(fact_1225_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none(nat) )
     => vEBT_VEBT_minNull(Ta) ) ).

% minminNull
tff(fact_1226_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1227_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1228_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_1229_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_1230_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_1231_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_1232_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_1233_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_1234_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1235_zdiv__zmult2__eq,axiom,
    ! [C2: int,A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_1236_split__zdiv,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_1237_zdiv__mono1,axiom,
    ! [A3: int,A7: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),A7)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A7),B2)) ) ) ).

% zdiv_mono1
tff(fact_1238_zdiv__mono2,axiom,
    ! [A3: int,B6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B6)) ) ) ) ).

% zdiv_mono2
tff(fact_1239_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1240_int__div__neg__eq,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1241_int__div__pos__eq,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1242_zdiv__mono1__neg,axiom,
    ! [A3: int,A7: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),A7)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A7),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_1243_zdiv__mono2__neg,axiom,
    ! [A3: int,B6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B6)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1244_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_1245_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
     => ( 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),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)) ) ) ).

% div_positive_int
tff(fact_1246_div__neg__pos__less0,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_1247_int__div__less__self,axiom,
    ! [Xb: int,K: 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)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),K)),Xb) ) ) ).

% int_div_less_self
tff(fact_1248_div__nonneg__neg__le0,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_1249_div__nonpos__pos__le0,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_1250_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A3) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1251_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1252_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1253_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1254_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1255_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))
      <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A3)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1256_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_1257_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_1258_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_1259_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_1260_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_1261_all__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),Nb)
         => aa(nat,$o,P,M5) )
    <=> ! [X: nat] :
          ( member(nat,X,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less
tff(fact_1262_ex__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),Nb)
          & aa(nat,$o,P,M5) )
    <=> ? [X: nat] :
          ( member(nat,X,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less
tff(fact_1263_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_1264_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1265_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_1266_realpow__pos__nth2,axiom,
    ! [A3: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ? [R3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),aa(nat,nat,suc,Nb)) = A3 ) ) ) ).

% realpow_pos_nth2
tff(fact_1267_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_1268_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((A3),(B2)),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_1269_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1270_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1271_realpow__pos__nth,axiom,
    ! [Nb: nat,A3: 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)),A3)
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),Nb) = A3 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1272_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A3: 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)),A3)
       => ? [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) = A3 )
            & ! [Y5: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y5)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y5),Nb) = A3 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1273_neg__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A3) ) ) ).

% neg_zdiv_mult_2
tff(fact_1274_pos__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A3) ) ) ).

% pos_zdiv_mult_2
tff(fact_1275_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1276_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_1277_int__power__div__base,axiom,
    ! [Mb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),K),Mb)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1278_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_1279_insertsimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(Ta)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insertsimp
tff(fact_1280_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_1281_member__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r2(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% member_bound_height'
tff(fact_1282_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [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 = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),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 = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),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 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                              $ite(
                                Xaa = Mi,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    Xaa = Ma,
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                            one_one(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),one_one(nat)) )) )) )) )) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_1283_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [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 = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),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 = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Mi,
                                zero_zero(nat),
                                $ite(
                                  Xaa = Ma,
                                  zero_zero(nat),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                    zero_zero(nat),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                      zero_zero(nat),
                                      $ite(
                                        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xaa)
                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma) ),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(nat)) ),
                                        zero_zero(nat) ) ) ) ) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_1284_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xb,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xb = vEBT_Leaf((A5),(B3))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [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 != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                            $let(
                              h: nat,
                              h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $ite(
                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                                & ~ ( ( Xaa = Mi )
                                    | ( Xaa = Ma ) ) ),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
                                one_one(nat) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_1285_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Mia,Xb),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
            $ite(
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
              & ~ ( ( Xb = Mia )
                  | ( Xb = Maa ) ) ),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)))),
                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_T_i_n_s_e_r_t(Summarya,h),one_one(nat))),
              one_one(nat) ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_1286_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))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) )
           => ( ! [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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_1287_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))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_1288_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))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S2)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_1289_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))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S2)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_1290_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1291_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_1292_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_1293_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_1294_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_1295_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: 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)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1296_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_1297_less__int__code_I1_J,axiom,
    ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_1298_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_1299_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_1300_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_1301_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_1302_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1303_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1304_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B6: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1305_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B6: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B6)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1306_q__pos__lemma,axiom,
    ! [B6: int,Q5: int,R4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_1307_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_1308_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
             => ( 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_1309_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_1310_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),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_1311_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_1312_set__bit__greater__eq,axiom,
    ! [K: int,Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K)) ).

% set_bit_greater_eq
tff(fact_1313_minNull__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_N_u_l_l(Ta)),one_one(nat)) ).

% minNull_bound
tff(fact_1314_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: $o] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf((Uu),$true)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_1315_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv: $o] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf($true,(Uv))) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_1316_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf($false,$false)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_1317_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : vEBT_T_m_i_n_N_u_l_l(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va,Vb,Vc)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_1318_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_T_m_i_n_N_u_l_l(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_1319_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xb) = Y )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => ( Y != one_one(nat) ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => ( Y != one_one(nat) ) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ( Y != one_one(nat) ) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)
                 => ( Y != one_one(nat) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_1320_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))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_1321_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))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S2) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,S2)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1322_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [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 = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,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 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                                    & ~ ( ( Xaa = Mi )
                                        | ( Xaa = Ma ) ) ),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                                      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
                                    one_one(nat) ) ) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_1323_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va2,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va2,Vb2)),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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_1324_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va2,Vb2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va2,Vb2)),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_1325_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xb,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xb = vEBT_Leaf((A5),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [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 != one_one(nat) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
                              one_one(nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_1326_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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,aa(nat,nat,aa(nat,fun(nat,nat),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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_1327_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_1328_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1329_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% atLeastatMost_empty_iff
tff(fact_1330_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_1331_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_1332_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_1333_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((A3),(B2)),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_1334_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,$o),ord_less_eq(nat),X4),M6) )
       => ~ ! [M2: nat] :
              ( aa(nat,$o,P,M2)
             => ~ ! [X5: nat] :
                    ( aa(nat,$o,P,X5)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),M2) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1335_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F3: fun(nat,fun(A,A)),A5: nat,B3: nat,Acc: A] : Xb != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A5),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1336_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero(nat),Ts,S),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_1337_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_1338_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeLista,Summarya),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_1339_insersimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ~ ? [X_12: nat] : vEBT_V8194947554948674370ptions(Ta,X_12)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,Y)),one_one(nat)) ) ) ).

% insersimp'
tff(fact_1340_insertsimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(Ta)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,L)),one_one(nat)) ) ) ).

% insertsimp'
tff(fact_1341_insert_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% insert'_bound_height
tff(fact_1342_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_1343_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_T_i_n_s_e_r_t2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_T_i_n_s_e_r_t2(Summarya,h),one_one(nat))),
            one_one(nat) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_1344_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [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 = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
                                  one_one(nat) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_1345_cpmi,axiom,
    ! [D5: int,P: fun(int,$o),P3: fun(int,$o),B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( ? [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),D5))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,B4)
                     => ( 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),D5)) ) )
         => ( ! [X4: int,K3: 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),K3),D5))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P3,X) )
                | ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa2: int] :
                        ( member(int,Xa2,B4)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_1346_cppi,axiom,
    ! [D5: int,P: fun(int,$o),P3: fun(int,$o),A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( ? [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),D5))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,A4)
                     => ( 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),D5)) ) )
         => ( ! [X4: int,K3: 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),K3),D5))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P3,X) )
                | ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa2: int] :
                        ( member(int,Xa2,A4)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa2),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_1347_bset_I6_J,axiom,
    ! [D5: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B4)
                 => ( X5 != 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),X5),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)),Ta) ) ) ) ).

% bset(6)
tff(fact_1348_bset_I8_J,axiom,
    ! [D5: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int)),B4)
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B4)
                   => ( X5 != 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),X5)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)) ) ) ) ) ).

% bset(8)
tff(fact_1349_aset_I6_J,axiom,
    ! [D5: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A4)
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A4)
                   => ( X5 != 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),X5),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_1350_aset_I8_J,axiom,
    ! [D5: int,A4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A4)
                 => ( X5 != 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),X5)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)) ) ) ) ).

% aset(8)
tff(fact_1351_bset_I3_J,axiom,
    ! [D5: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int)),B4)
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B4)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X5 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_1352_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,$o,Q6,X4) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
               => ( ( aa(A,$o,P,X5)
                    & aa(A,$o,Q,X5) )
                <=> ( aa(A,$o,P3,X5)
                    & aa(A,$o,Q6,X5) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_1353_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,$o,Q6,X4) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
               => ( ( aa(A,$o,P,X5)
                    | aa(A,$o,Q,X5) )
                <=> ( aa(A,$o,P3,X5)
                    | aa(A,$o,Q6,X5) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_1354_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( X5 != Ta ) ) ) ).

% pinf(3)
tff(fact_1355_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( X5 != Ta ) ) ) ).

% pinf(4)
tff(fact_1356_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Ta) ) ) ).

% pinf(5)
tff(fact_1357_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X5) ) ) ).

% pinf(7)
tff(fact_1358_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_1359_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,$o,Q6,X4) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
               => ( ( aa(A,$o,P,X5)
                    & aa(A,$o,Q,X5) )
                <=> ( aa(A,$o,P3,X5)
                    & aa(A,$o,Q6,X5) ) ) ) ) ) ) ).

% minf(1)
tff(fact_1360_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,$o,Q6,X4) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
               => ( ( aa(A,$o,P,X5)
                    | aa(A,$o,Q,X5) )
                <=> ( aa(A,$o,P3,X5)
                    | aa(A,$o,Q6,X5) ) ) ) ) ) ) ).

% minf(2)
tff(fact_1361_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( X5 != Ta ) ) ) ).

% minf(3)
tff(fact_1362_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( X5 != Ta ) ) ) ).

% minf(4)
tff(fact_1363_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Ta) ) ) ).

% minf(5)
tff(fact_1364_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X5) ) ) ).

% minf(7)
tff(fact_1365_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_1366_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X5) ) ) ).

% minf(8)
tff(fact_1367_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Ta) ) ) ).

% minf(6)
tff(fact_1368_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X5) ) ) ).

% pinf(8)
tff(fact_1369_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Ta) ) ) ).

% pinf(6)
tff(fact_1370_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D5: A,Q: fun(A,$o)] :
          ( ! [X4: A,K3: A] :
              ( aa(A,$o,P,X4)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D5))) )
         => ( ! [X4: A,K3: A] :
                ( aa(A,$o,Q,X4)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D5))) )
           => ! [X5: A,K4: A] :
                ( ( aa(A,$o,P,X5)
                  | aa(A,$o,Q,X5) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5)))
                  | aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1371_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D5: A,Q: fun(A,$o)] :
          ( ! [X4: A,K3: A] :
              ( aa(A,$o,P,X4)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D5))) )
         => ( ! [X4: A,K3: A] :
                ( aa(A,$o,Q,X4)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D5))) )
           => ! [X5: A,K4: A] :
                ( ( aa(A,$o,P,X5)
                  & aa(A,$o,Q,X5) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5)))
                  & aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1372_plusinfinity,axiom,
    ! [D2: int,P3: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X4: int,K3: 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),K3),D2))) )
       => ( ? [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_13: int] : aa(int,$o,P3,X_13)
           => ? [X_12: int] : aa(int,$o,P,X_12) ) ) ) ) ).

% plusinfinity
tff(fact_1373_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X4: int,K3: 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),K3),D2))) )
       => ( ? [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_13: int] : aa(int,$o,P1,X_13)
           => ? [X_12: int] : aa(int,$o,P,X_12) ) ) ) ) ).

% minusinfinity
tff(fact_1374_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X4: int] :
            ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X5: int] :
              ( aa(int,$o,P,X5)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1375_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X4: int] :
            ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X5: int] :
              ( aa(int,$o,P,X5)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1376_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X4: int,K3: int] :
            ( aa(int,$o,P,X4)
          <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D2))) )
       => ( ? [X_1: int] : aa(int,$o,P,X_1)
        <=> ? [X: int] :
              ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,P,X) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1377_aset_I7_J,axiom,
    ! [D5: int,A4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A4)
                 => ( X5 != 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),X5)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)) ) ) ) ).

% aset(7)
tff(fact_1378_aset_I5_J,axiom,
    ! [D5: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,A4)
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A4)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X5),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_1379_aset_I4_J,axiom,
    ! [D5: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,A4)
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A4)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X5 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_1380_aset_I3_J,axiom,
    ! [D5: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A4)
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A4)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X5 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_1381_bset_I7_J,axiom,
    ! [D5: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,B4)
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B4)
                   => ( X5 != 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),X5)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)) ) ) ) ) ).

% bset(7)
tff(fact_1382_bset_I5_J,axiom,
    ! [D5: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B4)
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X5),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)),Ta) ) ) ) ).

% bset(5)
tff(fact_1383_bset_I4_J,axiom,
    ! [D5: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,B4)
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B4)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X5 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_1384_psubsetI,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
     => ( ( A4 != B4 )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4) ) ) ).

% psubsetI
tff(fact_1385_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% unset_bit_0
tff(fact_1386_Bolzano,axiom,
    ! [A3: real,B2: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B2)
     => ( ! [A5: real,B3: real,C3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A5),B3)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B3),C3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A5),B3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),C3)
                 => aa(real,$o,aa(real,fun(real,$o),P,A5),C3) ) ) ) )
       => ( ! [X4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
               => ? [D6: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                    & ! [A5: real,B3: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A5),X4)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B3)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B3),A5)),D6) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A5),B3) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A3),B2) ) ) ) ).

% Bolzano
tff(fact_1387_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,Nb: nat] :
      ( ! [N: nat] : aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,H,N)
     => ( ( aa(nat,A,F2,zero_zero(nat)) = G )
       => ( aa(nat,A,F2,Nb) = $ite(Nb = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_1388_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_1389_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_1390_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_1391_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_1392_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_1393_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_1394_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_1395_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_1396_unset__bit__less__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K)),K) ).

% unset_bit_less_eq
tff(fact_1397_psubset__trans,axiom,
    ! [A: $tType,A4: set(A),B4: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B4),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),C5) ) ) ).

% psubset_trans
tff(fact_1398_psubsetD,axiom,
    ! [A: $tType,A4: set(A),B4: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
     => ( member(A,C2,A4)
       => member(A,C2,B4) ) ) ).

% psubsetD
tff(fact_1399_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
     => ? [B3: A] : member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1400_less__set__def,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A4)),aTP_Lamp_a(set(A),fun(A,$o),B4)) ) ).

% less_set_def
tff(fact_1401_not__psubset__empty,axiom,
    ! [A: $tType,A4: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_1402_psubsetE,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A4) ) ) ).

% psubsetE
tff(fact_1403_psubset__eq,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
        & ( A4 != B4 ) ) ) ).

% psubset_eq
tff(fact_1404_psubset__imp__subset,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4) ) ).

% psubset_imp_subset
tff(fact_1405_psubset__subset__trans,axiom,
    ! [A: $tType,A4: set(A),B4: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),C5) ) ) ).

% psubset_subset_trans
tff(fact_1406_subset__not__subset__eq,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
        & ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A4) ) ) ).

% subset_not_subset_eq
tff(fact_1407_subset__psubset__trans,axiom,
    ! [A: $tType,A4: set(A),B4: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B4),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),C5) ) ) ).

% subset_psubset_trans
tff(fact_1408_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
        | ( A4 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1409_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_1410_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_1411_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_1412_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R))
        <=> ( R = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_1413_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_1414_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_1415_obtain__set__pred,axiom,
    ! [Z: nat,Xb: nat,A4: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z),Xb)
     => ( vEBT_VEBT_min_in_set(A4,Z)
       => ( finite_finite(nat,A4)
         => ? [X_12: nat] : vEBT_is_pred_in_set(A4,Xb,X_12) ) ) ) ).

% obtain_set_pred
tff(fact_1416_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A4: set(nat),B4: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Z)
     => ( vEBT_VEBT_max_in_set(A4,Z)
       => ( finite_finite(nat,B4)
         => ( ( A4 = B4 )
           => ? [X_12: nat] : vEBT_is_succ_in_set(A4,Xb,X_12) ) ) ) ) ).

% obtain_set_succ
tff(fact_1417_prod__induct7,axiom,
    ! [G2: $tType,F: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))))] :
      ( ! [A5: A,B3: B,C3: C,D3: D,E2: E3,F3: F,G3: G2] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),B3),aa(product_prod(D,product_prod(E3,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),aa(C,fun(product_prod(D,product_prod(E3,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_Pair(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),C3),aa(product_prod(E3,product_prod(F,G2)),product_prod(D,product_prod(E3,product_prod(F,G2))),aa(D,fun(product_prod(E3,product_prod(F,G2)),product_prod(D,product_prod(E3,product_prod(F,G2)))),product_Pair(D,product_prod(E3,product_prod(F,G2))),D3),aa(product_prod(F,G2),product_prod(E3,product_prod(F,G2)),aa(E3,fun(product_prod(F,G2),product_prod(E3,product_prod(F,G2))),product_Pair(E3,product_prod(F,G2)),E2),aa(G2,product_prod(F,G2),aa(F,fun(G2,product_prod(F,G2)),product_Pair(F,G2),F3),G3)))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),$o,P,Xb) ) ).

% prod_induct7
tff(fact_1418_prod__induct6,axiom,
    ! [F: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))))] :
      ( ! [A5: A,B3: B,C3: C,D3: D,E2: E3,F3: F] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),A5),aa(product_prod(C,product_prod(D,product_prod(E3,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E3,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E3,F)))),B3),aa(product_prod(D,product_prod(E3,F)),product_prod(C,product_prod(D,product_prod(E3,F))),aa(C,fun(product_prod(D,product_prod(E3,F)),product_prod(C,product_prod(D,product_prod(E3,F)))),product_Pair(C,product_prod(D,product_prod(E3,F))),C3),aa(product_prod(E3,F),product_prod(D,product_prod(E3,F)),aa(D,fun(product_prod(E3,F),product_prod(D,product_prod(E3,F))),product_Pair(D,product_prod(E3,F)),D3),aa(F,product_prod(E3,F),aa(E3,fun(F,product_prod(E3,F)),product_Pair(E3,F),E2),F3))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),$o,P,Xb) ) ).

% prod_induct6
tff(fact_1419_prod__induct5,axiom,
    ! [E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3))))] :
      ( ! [A5: A,B3: B,C3: C,D3: D,E2: E3] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,E3))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E3))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E3)))),A5),aa(product_prod(C,product_prod(D,E3)),product_prod(B,product_prod(C,product_prod(D,E3))),aa(B,fun(product_prod(C,product_prod(D,E3)),product_prod(B,product_prod(C,product_prod(D,E3)))),product_Pair(B,product_prod(C,product_prod(D,E3))),B3),aa(product_prod(D,E3),product_prod(C,product_prod(D,E3)),aa(C,fun(product_prod(D,E3),product_prod(C,product_prod(D,E3))),product_Pair(C,product_prod(D,E3)),C3),aa(E3,product_prod(D,E3),aa(D,fun(E3,product_prod(D,E3)),product_Pair(D,E3),D3),E2)))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),$o,P,Xb) ) ).

% prod_induct5
tff(fact_1420_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => finite_finite(nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_1421_succ__none__empty,axiom,
    ! [Xs: set(nat),A3: nat] :
      ( ~ ? [X_12: nat] : vEBT_is_succ_in_set(Xs,A3,X_12)
     => ( finite_finite(nat,Xs)
       => ~ ? [X5: nat] :
              ( member(nat,X5,Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),X5) ) ) ) ).

% succ_none_empty
tff(fact_1422_pred__none__empty,axiom,
    ! [Xs: set(nat),A3: nat] :
      ( ~ ? [X_12: nat] : vEBT_is_pred_in_set(Xs,A3,X_12)
     => ( finite_finite(nat,Xs)
       => ~ ? [X5: nat] :
              ( member(nat,X5,Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X5),A3) ) ) ) ).

% pred_none_empty
tff(fact_1423_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_1424_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A7: A,B6: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B6) )
    <=> ( ( A3 = A7 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
tff(fact_1425_mod__mod__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,modulo_modulo(A,A3,B2),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mod_trivial
tff(fact_1426_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ).

% mod_0
tff(fact_1427_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,zero_zero(A)) = A3 ) ).

% mod_by_0
tff(fact_1428_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,A3) = zero_zero(A) ) ).

% mod_self
tff(fact_1429_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_1430_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_add_self1
tff(fact_1431_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_add_self2
tff(fact_1432_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2),B2) = modulo_modulo(A,A3,B2) ) ).

% minus_mod_self2
tff(fact_1433_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_1434_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),B2) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_1435_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),B2) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_1436_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_1437_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_1438_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A3,B2)),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_1439_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A3,B2)),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_1440_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self1
tff(fact_1441_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self2
tff(fact_1442_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self3
tff(fact_1443_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self4
tff(fact_1444_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ finite_finite(A,set_or1337092689740270186AtMost(A,A3,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% infinite_Icc_iff
tff(fact_1445_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_1446_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self1
tff(fact_1447_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self2
tff(fact_1448_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,Mb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self3
tff(fact_1449_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,Mb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self4
tff(fact_1450_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_1451_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_1452_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_1453_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),Nb)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1454_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1455_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1456_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_1457_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_1458_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_1459_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_1460_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_1461_unique__remainder,axiom,
    ! [A3: int,B2: int,Q2: int,R: int,Q5: int,R4: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R4))
       => ( R = R4 ) ) ) ).

% unique_remainder
tff(fact_1462_unique__quotient,axiom,
    ! [A3: int,B2: int,Q2: int,R: int,Q5: int,R4: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R4))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_1463_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).

% mod_mult_eq
tff(fact_1464_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,A7: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A7,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A7),B6),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_1465_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,B2)),C2) ) ).

% mod_mult_mult2
tff(fact_1466_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A3,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% mult_mod_right
tff(fact_1467_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).

% mod_mult_left_eq
tff(fact_1468_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).

% mod_mult_right_eq
tff(fact_1469_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C2) ) ).

% mod_add_eq
tff(fact_1470_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,A7: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A7,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A7),B6),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_1471_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C2) ) ).

% mod_add_left_eq
tff(fact_1472_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C2) ) ).

% mod_add_right_eq
tff(fact_1473_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2),C2) ) ).

% mod_diff_eq
tff(fact_1474_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,A7: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A7,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A7),B6),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_1475_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2),C2) ) ).

% mod_diff_left_eq
tff(fact_1476_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2),C2) ) ).

% mod_diff_right_eq
tff(fact_1477_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,Nb: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),modulo_modulo(A,A3,B2)),Nb),B2) = modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb),B2) ) ).

% power_mod
tff(fact_1478_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_1479_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_1480_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_1481_finite__nat__set__iff__bounded,axiom,
    ! [N2: set(nat)] :
      ( finite_finite(nat,N2)
    <=> ? [M5: nat] :
        ! [X: nat] :
          ( member(nat,X,N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1482_bounded__nat__set__is__finite,axiom,
    ! [N2: set(nat),Nb: nat] :
      ( ! [X4: nat] :
          ( member(nat,X4,N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),Nb) )
     => finite_finite(nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_1483_finite__nat__set__iff__bounded__le,axiom,
    ! [N2: set(nat)] :
      ( finite_finite(nat,N2)
    <=> ? [M5: nat] :
        ! [X: nat] :
          ( member(nat,X,N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1484_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : finite_finite(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_1485_finite__less__ub,axiom,
    ! [F2: fun(nat,nat),U: nat] :
      ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,F2,N))
     => finite_finite(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ag(fun(nat,nat),fun(nat,fun(nat,$o)),F2),U))) ) ).

% finite_less_ub
tff(fact_1486_finite__lists__length__eq,axiom,
    ! [A: $tType,A4: set(A),Nb: nat] :
      ( finite_finite(A,A4)
     => finite_finite(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ah(set(A),fun(nat,fun(list(A),$o)),A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_1487_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_1488_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_1489_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),modulo_modulo(A,A3,B2)),A3) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1490_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$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,A3,B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1491_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1492_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = A3 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1493_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_1494_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ).

% mod_eqE
tff(fact_1495_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_add1_eq
tff(fact_1496_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_1497_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_1498_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_1499_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_1500_mod__eq__0D,axiom,
    ! [Mb: nat,D2: nat] :
      ( ( modulo_modulo(nat,Mb,D2) = zero_zero(nat) )
     => ? [Q3: nat] : Mb = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q3) ) ).

% mod_eq_0D
tff(fact_1501_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_1502_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_1503_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_1504_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_1505_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ finite_finite(A,set_or1337092689740270186AtMost(A,A3,B2)) ) ) ).

% infinite_Icc
tff(fact_1506_finite__lists__length__le,axiom,
    ! [A: $tType,A4: set(A),Nb: nat] :
      ( finite_finite(A,A4)
     => finite_finite(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ai(set(A),fun(nat,fun(list(A),$o)),A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_1507_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( modulo_modulo(A,A3,B2) = A3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1508_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$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,A3,B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1509_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1510_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_1511_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(8)
tff(fact_1512_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(6)
tff(fact_1513_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_mult1_eq
tff(fact_1514_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))),modulo_modulo(A,A3,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_1515_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_1516_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2)) ) ).

% mod_div_decomp
tff(fact_1517_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2)) = A3 ) ).

% div_mult_mod_eq
tff(fact_1518_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)) = A3 ) ).

% mod_div_mult_eq
tff(fact_1519_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) = A3 ) ).

% mod_mult_div_eq
tff(fact_1520_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))),modulo_modulo(A,A3,B2)) = A3 ) ).

% mult_div_mod_eq
tff(fact_1521_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) = modulo_modulo(A,A3,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_1522_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_1523_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_1524_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)) = modulo_modulo(A,A3,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_1525_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_1526_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(12)
tff(fact_1527_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(10)
tff(fact_1528_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_1529_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Mb,Q2) = modulo_modulo(nat,Nb,Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
       => ~ ! [S2: nat] : Mb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_1530_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Mb,Q2) = modulo_modulo(nat,Nb,Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ~ ! [S2: nat] : Nb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_1531_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xb,Nb) = modulo_modulo(nat,Y,Nb) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xb)
       => ? [Q3: nat] : Xb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1532_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_1533_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : modulo_modulo(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb),Q2))),modulo_modulo(nat,Mb,Nb)) ).

% mod_mult2_eq
tff(fact_1534_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique8689654367752047608divmod(nat,Mb,Nb) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(num,nat,numeral_numeral(nat),Mb)),aa(num,nat,numeral_numeral(nat),Nb))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),Mb),aa(num,nat,numeral_numeral(nat),Nb))) ).

% divmod'_nat_def
tff(fact_1535_modulo__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : modulo_modulo(nat,Mb,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_1536_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A5: A,B3: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) ).

% old.prod.exhaust
tff(fact_1537_surj__pair,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B)] :
    ? [X4: A,Y4: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) ).

% surj_pair
tff(fact_1538_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o),P2: product_prod(A,B)] :
      ( ! [A5: A,B3: B] : aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3))
     => aa(product_prod(A,B),$o,P,P2) ) ).

% prod_cases
tff(fact_1539_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A7: A,B6: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B6) )
     => ~ ( ( A3 = A7 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
tff(fact_1540_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_1541_split__mod,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,modulo_modulo(nat,Mb,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,Mb) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I4)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_1542_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : unique8689654367752047608divmod(A,Mb,Nb) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb))),modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Nb))) ) ).

% divmod_def
tff(fact_1543_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),C2))),modulo_modulo(A,A3,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1544_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_1545_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_1546_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_1547_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_1548_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$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,A3,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,A3,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,A3,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1549_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A3 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_1550_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat,Mb: nat] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A3,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_1551_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$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,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1552_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat,A3: A] :
          ( aa(nat,$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),A3),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,A3,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_1553_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A5: A,B3: B,C3: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)) ).

% prod_cases3
tff(fact_1554_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))] :
      ( ! [A5: A,B3: B,C3: C] : aa(product_prod(A,product_prod(B,C)),$o,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)))
     => aa(product_prod(A,product_prod(B,C)),$o,P,Xb) ) ).

% prod_induct3
tff(fact_1555_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int)) ),
              Q2 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1556_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_1557_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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,A3,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,A3,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,A3,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1558_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,Nb)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% unset_bit_Suc
tff(fact_1559_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,Nb)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% set_bit_Suc
tff(fact_1560_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1561_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_1562_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A5: A,B3: B,C3: C,D3: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D3))) ).

% prod_cases4
tff(fact_1563_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3))))] :
      ~ ! [A5: A,B3: B,C3: C,D3: D,E2: E3] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E3))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E3))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E3)))),A5),aa(product_prod(C,product_prod(D,E3)),product_prod(B,product_prod(C,product_prod(D,E3))),aa(B,fun(product_prod(C,product_prod(D,E3)),product_prod(B,product_prod(C,product_prod(D,E3)))),product_Pair(B,product_prod(C,product_prod(D,E3))),B3),aa(product_prod(D,E3),product_prod(C,product_prod(D,E3)),aa(C,fun(product_prod(D,E3),product_prod(C,product_prod(D,E3))),product_Pair(C,product_prod(D,E3)),C3),aa(E3,product_prod(D,E3),aa(D,fun(E3,product_prod(D,E3)),product_Pair(D,E3),D3),E2)))) ).

% prod_cases5
tff(fact_1564_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))))] :
      ~ ! [A5: A,B3: B,C3: C,D3: D,E2: E3,F3: F] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),A5),aa(product_prod(C,product_prod(D,product_prod(E3,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E3,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E3,F)))),B3),aa(product_prod(D,product_prod(E3,F)),product_prod(C,product_prod(D,product_prod(E3,F))),aa(C,fun(product_prod(D,product_prod(E3,F)),product_prod(C,product_prod(D,product_prod(E3,F)))),product_Pair(C,product_prod(D,product_prod(E3,F))),C3),aa(product_prod(E3,F),product_prod(D,product_prod(E3,F)),aa(D,fun(product_prod(E3,F),product_prod(D,product_prod(E3,F))),product_Pair(D,product_prod(E3,F)),D3),aa(F,product_prod(E3,F),aa(E3,fun(F,product_prod(E3,F)),product_Pair(E3,F),E2),F3))))) ).

% prod_cases6
tff(fact_1565_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))))] :
      ~ ! [A5: A,B3: B,C3: C,D3: D,E2: E3,F3: F,G3: G2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),B3),aa(product_prod(D,product_prod(E3,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),aa(C,fun(product_prod(D,product_prod(E3,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_Pair(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),C3),aa(product_prod(E3,product_prod(F,G2)),product_prod(D,product_prod(E3,product_prod(F,G2))),aa(D,fun(product_prod(E3,product_prod(F,G2)),product_prod(D,product_prod(E3,product_prod(F,G2)))),product_Pair(D,product_prod(E3,product_prod(F,G2))),D3),aa(product_prod(F,G2),product_prod(E3,product_prod(F,G2)),aa(E3,fun(product_prod(F,G2),product_prod(E3,product_prod(F,G2))),product_Pair(E3,product_prod(F,G2)),E2),aa(G2,product_prod(F,G2),aa(F,fun(G2,product_prod(F,G2)),product_Pair(F,G2),F3),G3)))))) ).

% prod_cases7
tff(fact_1566_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)))] :
      ( ! [A5: A,B3: B,C3: C,D3: D] : aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D3))))
     => aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,Xb) ) ).

% prod_induct4
tff(fact_1567_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),Nb)),
                $ite(modulo_modulo(nat,B4,Nb) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_1568_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_1569_finite__Collect__le__nat,axiom,
    ! [K: nat] : finite_finite(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aj(nat,fun(nat,$o)),K))) ).

% finite_Collect_le_nat
tff(fact_1570_finite__Collect__less__nat,axiom,
    ! [K: nat] : finite_finite(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),K))) ).

% finite_Collect_less_nat
tff(fact_1571_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => finite_finite(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_al(nat,fun(A,$o),Nb))) ) ) ).

% finite_roots_unity
tff(fact_1572_div__mod__decomp,axiom,
    ! [A4: nat,Nb: nat] : A4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),Nb)),Nb)),modulo_modulo(nat,A4,Nb)) ).

% div_mod_decomp
tff(fact_1573_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,Nb),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se8732182000553998342ip_bit(A,Nb,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% flip_bit_Suc
tff(fact_1574_div__less__mono,axiom,
    ! [A4: nat,B4: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( modulo_modulo(nat,A4,Nb) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B4,Nb) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_1575_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( aa(num,num,bit0,X22) = aa(num,num,bit0,Y2) )
    <=> ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_1576_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa(num,num,bit1,X32) = aa(num,num,bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_1577_finite__interval__int4,axiom,
    ! [A3: int,B2: int] : finite_finite(int,aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_am(int,fun(int,fun(int,$o)),A3),B2))) ).

% finite_interval_int4
tff(fact_1578_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_1579_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_1580_finite__interval__int3,axiom,
    ! [A3: int,B2: int] : finite_finite(int,aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_an(int,fun(int,fun(int,$o)),A3),B2))) ).

% finite_interval_int3
tff(fact_1581_finite__interval__int2,axiom,
    ! [A3: int,B2: int] : finite_finite(int,aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ao(int,fun(int,fun(int,$o)),A3),B2))) ).

% finite_interval_int2
tff(fact_1582_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1583_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1584_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_1585_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_1586_zmod__le__nonneg__dividend,axiom,
    ! [Mb: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Mb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,Mb,K)),Mb) ) ).

% zmod_le_nonneg_dividend
tff(fact_1587_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,K,L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1588_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),modulo_modulo(int,K,L)) ) ).

% neg_mod_bound
tff(fact_1589_zmod__eq__0__iff,axiom,
    ! [Mb: int,D2: int] :
      ( ( modulo_modulo(int,Mb,D2) = zero_zero(int) )
    <=> ? [Q4: int] : Mb = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q4) ) ).

% zmod_eq_0_iff
tff(fact_1590_zmod__eq__0D,axiom,
    ! [Mb: int,D2: int] :
      ( ( modulo_modulo(int,Mb,D2) = zero_zero(int) )
     => ? [Q3: int] : Mb = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q3) ) ).

% zmod_eq_0D
tff(fact_1591_zdiv__mono__strict,axiom,
    ! [A4: int,B4: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A4),B4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => ( ( modulo_modulo(int,A4,Nb) = zero_zero(int) )
         => ( ( modulo_modulo(int,B4,Nb) = zero_zero(int) )
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),Nb)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1592_finite__maxlen,axiom,
    ! [A: $tType,M6: set(list(A))] :
      ( finite_finite(list(A),M6)
     => ? [N: nat] :
        ! [X5: list(A)] :
          ( member(list(A),X5,M6)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X5)),N) ) ) ).

% finite_maxlen
tff(fact_1593_div__mod__decomp__int,axiom,
    ! [A4: int,Nb: int] : A4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),Nb)),Nb)),modulo_modulo(int,A4,Nb)) ).

% div_mod_decomp_int
tff(fact_1594_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( modulo_modulo(int,K,L) = R ) ) ).

% mod_int_unique
tff(fact_1595_neg__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,A3,B2)),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),modulo_modulo(int,A3,B2)) ) ) ).

% neg_mod_conj
tff(fact_1596_pos__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A3,B2))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,A3,B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_1597_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( modulo_modulo(int,I,K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_1598_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1599_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,K,L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_1600_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% verit_la_disequality
tff(fact_1601_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),A3) ) ).

% verit_comp_simplify1(2)
tff(fact_1602_divmod__int__def,axiom,
    ! [Mb: num,Nb: num] : unique8689654367752047608divmod(int,Mb,Nb) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),Mb)),aa(num,int,numeral_numeral(int),Nb))),modulo_modulo(int,aa(num,int,numeral_numeral(int),Mb),aa(num,int,numeral_numeral(int),Nb))) ).

% divmod_int_def
tff(fact_1603_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),A3) ) ).

% verit_comp_simplify1(1)
tff(fact_1604_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)),modulo_modulo(int,K,L))) ).

% eucl_rel_int
tff(fact_1605_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1606_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1607_verit__le__mono__div__int,axiom,
    ! [A4: int,B4: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A4),B4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => aa(int,$o,
            aa(int,fun(int,$o),ord_less_eq(int),
              aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),Nb)),
                $ite(modulo_modulo(int,B4,Nb) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_1608_split__zmod,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,modulo_modulo(int,Nb,K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,Nb) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_1609_int__mod__neg__eq,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R)
         => ( modulo_modulo(int,A3,B2) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1610_int__mod__pos__eq,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( modulo_modulo(int,A3,B2) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1611_zmod__zmult2__eq,axiom,
    ! [C2: int,A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( modulo_modulo(int,A3,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2),C2))),modulo_modulo(int,A3,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1612_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K)),modulo_modulo(int,Nb,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1613_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K)),modulo_modulo(int,Nb,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1614_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B6: A,A7: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B6),A7)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A7),B6) ) ) ).

% verit_comp_simplify1(3)
tff(fact_1615_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% verit_sum_simplify
tff(fact_1616_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one2 != aa(num,num,bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_1617_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite(A,A4)
         => ( member(A,A3,A4)
           => ? [X4: A] :
                ( member(A,X4,A4)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X4)
                & ! [Xa: A] :
                    ( member(A,Xa,A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1618_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite(A,A4)
         => ( member(A,A3,A4)
           => ? [X4: A] :
                ( member(A,X4,A4)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A3)
                & ! [Xa: A] :
                    ( member(A,Xa,A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X4)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1619_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : aa(num,num,bit0,X22) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_1620_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_1621_finite__psubset__induct,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),$o)] :
      ( finite_finite(A,A4)
     => ( ! [A8: set(A)] :
            ( finite_finite(A,A8)
           => ( ! [B7: set(A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B7),A8)
                 => aa(set(A),$o,P,B7) )
             => aa(set(A),$o,P,A8) ) )
       => aa(set(A),$o,P,A4) ) ) ).

% finite_psubset_induct
tff(fact_1622_pos__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( 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))),A3)) = 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,A3))) ) ) ).

% pos_zmod_mult_2
tff(fact_1623_neg__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),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))),A3)) = 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)),A3))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_1624_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A)] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,A4)
                & ! [Xa: A] :
                    ( member(A,Xa,A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X4)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1625_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A)] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,A4)
                & ! [Xa: A] :
                    ( member(A,Xa,A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1626_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1627_finite__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => finite_finite(complex,aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_ap(nat,fun(complex,fun(complex,$o)),Nb),C2))) ) ).

% finite_nth_roots
tff(fact_1628_product__nth,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),Nb) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,Nb,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_1629_old_Oprod_Orec,axiom,
    ! [B: $tType,A: $tType,C: $tType,F1: fun(B,fun(C,A)),A3: 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),A3),B2)) = aa(C,A,aa(B,fun(C,A),F1,A3),B2) ).

% old.prod.rec
tff(fact_1630_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_aq(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_aq(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite(A,aa(fun(A,$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_ar(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_1631_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_as(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_as(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite(A,aa(fun(A,$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_at(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_1632_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_1633_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A3: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Leaf((A3),(B2))) = zero_zero(nat) ).

% VEBT_internal.height.simps(1)
tff(fact_1634_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_1635_dbl__inc__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,K)) ) ).

% dbl_inc_simps(5)
tff(fact_1636_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_1637_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_1638_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A5: $o,B3: $o] : Xb != vEBT_Leaf((A5),(B3))
     => ~ ! [Uu2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb != vEBT_Node(Uu2,Deg,TreeList,Summary) ) ).

% VEBT_internal.height.cases
tff(fact_1639_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_1640_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mia),Maa))),Dega,list_update(vEBT_VEBT,TreeLista,vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_1641_insert__simp__norm,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xb),Maa))),Dega,list_update(vEBT_VEBT,TreeLista,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_1642_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Mb: nat,Nb: nat] :
      ( ! [M2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M2),zero_zero(nat))
     => ( ! [M2: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N),modulo_modulo(nat,M2,N))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M2),N) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_1643_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,Nb),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,bit_concat_bit(Nb,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),L))) ).

% concat_bit_Suc
tff(fact_1644_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_1645_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_1646_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_1647_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_1648_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_1649_max__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),zero_zero(nat)) = A3 ).

% max_nat.right_neutral
tff(fact_1650_max__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1651_max__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A3) = A3 ).

% max_nat.left_neutral
tff(fact_1652_max__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_1653_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_1654_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_1655_concat__bit__0,axiom,
    ! [K: int,L: int] : aa(int,int,bit_concat_bit(zero_zero(nat),K),L) = L ).

% concat_bit_0
tff(fact_1656_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_1657_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_1658_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_1659_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_1660_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_1661_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_1662_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_1663_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_1664_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_1665_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_concat_bit(Nb,K),L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_1666_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_concat_bit(Nb,K),L)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_1667_dbl__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,K)) ) ).

% dbl_simps(5)
tff(fact_1668_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_1669_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_1670_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_1671_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_1672_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_1673_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q2)) ).

% nat_add_max_right
tff(fact_1674_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),Nb)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Q2)) ).

% nat_add_max_left
tff(fact_1675_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)) ).

% nat_mult_max_right
tff(fact_1676_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),Nb)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ).

% nat_mult_max_left
tff(fact_1677_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_1678_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X5),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Xa),Xa,X5) ) ).

% max_def_raw
tff(fact_1679_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,Mb: nat,L: int,R: int] : aa(int,int,bit_concat_bit(Nb,K),aa(int,int,bit_concat_bit(Mb,L),R)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb),aa(int,int,bit_concat_bit(Nb,K),L)),R) ).

% concat_bit_assoc
tff(fact_1680_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_1681_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_1682_semiring__norm_I27_J,axiom,
    ! [Nb: num] : bitM(aa(num,num,bit0,Nb)) = aa(num,num,bit1,bitM(Nb)) ).

% semiring_norm(27)
tff(fact_1683_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_1684_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_1685_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_1686_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_1687_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_1688_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_1689_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_1690_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_1691_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A3: nat,B2: nat] :
      ( ! [A5: nat,B3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A5),B3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B3),A5) )
     => ( ! [A5: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A5),zero_zero(nat))
       => ( ! [A5: nat,B3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A5),B3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),B3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A3),B2) ) ) ) ).

% Euclid_induct
tff(fact_1692_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Xb,Mia)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Maa))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeLista,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_1693_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A5),$true),vEBT_Leaf((A5),(B3))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
             => ( Y != vEBT_Node(Info2,zero_zero(nat),Ts2,S2) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
               => ( Y != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) ) )
           => ( ! [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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),
                                    aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Xaa,Mi)),
                                    aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_1694_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))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A5),$true),vEBT_Leaf((A5),(B3))) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
               => ( ( Y = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                      aa(nat,product_prod(nat,nat),
                                        aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Xaa,Mi)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_1695_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb3
tff(fact_1696_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_1697_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_1698_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb1
tff(fact_1699_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_1700_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% max.bounded_iff
tff(fact_1701_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),zero_zero(extended_enat)),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_1702_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Q2),zero_zero(extended_enat)) = Q2 ).

% max_enat_simps(2)
tff(fact_1703_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod(nat,nat)] :
      ~ ! [K3: nat,M2: nat] : Xb != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K3),M2) ).

% prod_decode_aux.cases
tff(fact_1704_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ) ).

% max.mono
tff(fact_1705_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.orderE
tff(fact_1706_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% max.orderI
tff(fact_1707_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% max.boundedE
tff(fact_1708_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3) ) ) ) ).

% max.boundedI
tff(fact_1709_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.order_iff
tff(fact_1710_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ).

% max.cobounded1
tff(fact_1711_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ).

% max.cobounded2
tff(fact_1712_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_1713_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb_iff1
tff(fact_1714_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_1715_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).

% max.coboundedI1
tff(fact_1716_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).

% max.coboundedI2
tff(fact_1717_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_1718_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_1719_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_1720_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% max.strict_boundedE
tff(fact_1721_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_1722_triangle__def,axiom,
    ! [Nb: nat] : nat_triangle(Nb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% triangle_def
tff(fact_1723_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Nb: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ? [Y5: A] :
                ( aa(A,$o,P,Y5)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y5)),aa(A,nat,F2,X4)) ) )
       => ? [Y4: A] :
            ( aa(A,$o,P,Y4)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),Nb)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1724_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( aa(nat,$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)),A3),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,A3,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_1725_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_au(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Mb,Nb)) ) ).

% divmod_algorithm_code(6)
tff(fact_1726_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),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_1727_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X22: A] : aa(option(A),nat,size_option(A,Xb),aa(A,option(A),some(A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_1728_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Nb)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_1729_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_1730_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( dvd_dvd(A,zero_zero(A),A3)
        <=> ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_1731_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : dvd_dvd(A,A3,zero_zero(A)) ) ).

% dvd_0_right
tff(fact_1732_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_1733_dvd__1__left,axiom,
    ! [K: nat] : dvd_dvd(nat,aa(nat,nat,suc,zero_zero(nat)),K) ).

% dvd_1_left
tff(fact_1734_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3))
        <=> dvd_dvd(A,A3,B2) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1735_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))
        <=> dvd_dvd(A,A3,B2) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1736_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( dvd_dvd(A,A3,C2)
           => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3))
            <=> dvd_dvd(A,B2,C2) ) ) ) ) ).

% div_dvd_div
tff(fact_1737_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( ( K = zero_zero(nat) )
        | dvd_dvd(nat,Mb,Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1738_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_1739_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F2,A3),B2) ).

% case_prod_conv
tff(fact_1740_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3))
          <=> dvd_dvd(A,B2,C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1741_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))
          <=> dvd_dvd(A,B2,C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1742_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | dvd_dvd(A,A3,B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1743_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( C2 = zero_zero(A) )
            | dvd_dvd(A,A3,B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1744_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( dvd_dvd(A,B2,one_one(A))
           => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_1745_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)))
        <=> dvd_dvd(A,A3,B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1746_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B2))
        <=> dvd_dvd(A,A3,B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1747_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_1748_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),A3) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_1749_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = A3 ) ) ) ).

% unit_div_1_div_1
tff(fact_1750_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_1751_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( dvd_dvd(A,B2,one_one(A))
           => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),one_one(A)) ) ) ) ).

% unit_div
tff(fact_1752_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( dvd_dvd(A,C2,A3)
         => ( dvd_dvd(A,C2,B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_add
tff(fact_1753_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A3: A,B2: A] :
          ( dvd_dvd(A,C2,A3)
         => ( dvd_dvd(A,C2,B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_diff
tff(fact_1754_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( modulo_modulo(A,B2,A3) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1755_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_1756_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_1757_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_1758_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_1759_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_1760_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),A3) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_1761_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) ) ) ) ).

% unit_mult_div_div
tff(fact_1762_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Nb: nat,A3: A,B2: A] :
          ( aa(nat,$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),A3),Nb),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
          <=> dvd_dvd(A,A3,B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_1763_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: 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),A3),B2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_mult_iff
tff(fact_1764_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: 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),A3),B2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
          <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_add
tff(fact_1765_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: 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),A3),B2))
        <=> ~ ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
            <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% odd_add
tff(fact_1766_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),modulo_modulo(A,A3,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)),A3) ) ) ).

% even_mod_2_iff
tff(fact_1767_even__Suc__div__two,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),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ).

% even_Suc_div_two
tff(fact_1768_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_1769_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_1770_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Nb))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,Nb,Mb)) ) ) ).

% dvd_numeral_simp
tff(fact_1771_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1772_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1773_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),zero_zero(A))
        <=> ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_1774_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: 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),A3),one_one(A)))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ).

% even_plus_one_iff
tff(fact_1775_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: 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),A3),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),A3),B2)) ) ) ).

% even_diff
tff(fact_1776_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_1777_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_1778_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
              & ( A3 != zero_zero(A) ) )
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1779_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1780_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1781_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1782_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% even_power
tff(fact_1783_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_1784_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,Mb),aa(num,num,bit0,Nb)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_av(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Mb,Nb)) ) ).

% divmod_algorithm_code(5)
tff(fact_1785_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),one_one(A)) = A3 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1786_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( aa(nat,$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),A3),zero_zero(A)) )
              | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1787_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)),one_one(A)))
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1788_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_1789_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( dvd_dvd(A,B2,C2)
           => dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_trans
tff(fact_1790_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : dvd_dvd(A,A3,A3) ) ).

% dvd_refl
tff(fact_1791_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( dvd_dvd(A,zero_zero(A),A3)
         => ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1792_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,B2)
        <=> ( ( A3 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_1793_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ? [B8: A,C6: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B8),C6) )
              & dvd_dvd(A,B8,B2)
              & dvd_dvd(A,C6,C2) ) ) ) ).

% division_decomp
tff(fact_1794_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P2: A,A3: A,B2: A] :
          ( dvd_dvd(A,P2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
         => ~ ! [X4: A,Y4: A] :
                ( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X4),Y4) )
               => ( dvd_dvd(A,X4,A3)
                 => ~ dvd_dvd(A,Y4,B2) ) ) ) ) ).

% dvd_productE
tff(fact_1795_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) ) ).

% dvd_triv_right
tff(fact_1796_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
         => dvd_dvd(A,B2,C2) ) ) ).

% dvd_mult_right
tff(fact_1797_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( dvd_dvd(A,C2,D2)
           => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ).

% mult_dvd_mono
tff(fact_1798_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% dvd_triv_left
tff(fact_1799_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
         => dvd_dvd(A,A3,C2) ) ) ).

% dvd_mult_left
tff(fact_1800_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,B2)
         => dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult2
tff(fact_1801_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,C2: A,B2: A] :
          ( dvd_dvd(A,A3,C2)
         => dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult
tff(fact_1802_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( dvd_dvd(A,B2,A3)
        <=> ? [K2: A] : A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvd_def
tff(fact_1803_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A3: A,B2: A,K: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
         => dvd_dvd(A,B2,A3) ) ) ).

% dvdI
tff(fact_1804_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( dvd_dvd(A,B2,A3)
         => ~ ! [K3: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvdE
tff(fact_1805_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( dvd_dvd(A,B2,one_one(A))
           => dvd_dvd(A,A3,one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1806_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => dvd_dvd(A,B2,A3) ) ) ).

% unit_imp_dvd
tff(fact_1807_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : dvd_dvd(A,one_one(A),A3) ) ).

% one_dvd
tff(fact_1808_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_1809_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( dvd_dvd(A,A3,C2)
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> dvd_dvd(A,A3,B2) ) ) ) ).

% dvd_add_left_iff
tff(fact_1810_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( dvd_dvd(A,A3,C2)
           => dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ) ).

% dvd_add
tff(fact_1811_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A3: A,C2: A,B2: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% dvd_diff_commute
tff(fact_1812_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_1813_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A3: A,B2: A] :
          ( dvd_dvd(A,C2,A3)
         => ( dvd_dvd(A,C2,B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
            <=> ( A3 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1814_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
         => ( dvd_dvd(A,C2,A3)
           => ( dvd_dvd(A,C2,B2)
             => ( A3 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1815_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,B2: A,A3: A] :
          ( dvd_dvd(A,D2,B2)
         => ( dvd_dvd(A,B2,A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).

% div_div_div_same
tff(fact_1816_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_1817_old_Oprod_Ocase,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),X1: B,X22: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),X22)) = aa(C,A,aa(B,fun(C,A),F2,X1),X22) ).

% old.prod.case
tff(fact_1818_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A3: A] :
          ( dvd_dvd(A,C2,B2)
         => ( modulo_modulo(A,modulo_modulo(A,A3,B2),C2) = modulo_modulo(A,A3,C2) ) ) ) ).

% mod_mod_cancel
tff(fact_1819_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K: A,Mb: A,Nb: A] :
          ( dvd_dvd(A,K,Mb)
         => ( dvd_dvd(A,K,Nb)
           => dvd_dvd(A,K,modulo_modulo(A,Mb,Nb)) ) ) ) ).

% dvd_mod
tff(fact_1820_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A3: A,B2: A] :
          ( dvd_dvd(A,C2,modulo_modulo(A,A3,B2))
         => ( dvd_dvd(A,C2,B2)
           => dvd_dvd(A,C2,A3) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1821_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B2: A,A3: A] :
          ( dvd_dvd(A,C2,B2)
         => ( dvd_dvd(A,C2,modulo_modulo(A,A3,B2))
          <=> dvd_dvd(A,C2,A3) ) ) ) ).

% dvd_mod_iff
tff(fact_1822_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% signed_take_bit_mult
tff(fact_1823_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_1824_dvd__diff__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,K,Mb)
     => ( dvd_dvd(nat,K,Nb)
       => dvd_dvd(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_1825_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% signed_take_bit_diff
tff(fact_1826_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aw(A,fun(A,$o),A3))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aw(A,fun(A,$o),B2)))
        <=> dvd_dvd(A,A3,B2) ) ) ).

% subset_divisors_dvd
tff(fact_1827_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X4: A,Y4: B] : aa(B,C,aa(A,fun(B,C),F2,X4),Y4) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2) = G ) ) ).

% cond_case_prod_eta
tff(fact_1828_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_ax(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2)) = F2 ).

% case_prod_eta
tff(fact_1829_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,$o,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z))
     => ~ ! [X4: B,Y4: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y4) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_1830_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aw(A,fun(A,$o),A3))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aw(A,fun(A,$o),B2)))
        <=> ( dvd_dvd(A,A3,B2)
            & ~ dvd_dvd(A,B2,A3) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1831_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_ri4674362597316999326ke_bit(A,Mb),A3))
        <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ).

% even_signed_take_bit_iff
tff(fact_1832_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_1833_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),S))
          <=> dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),S)) ) ) ) ).

% pinf(9)
tff(fact_1834_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),S))
          <=> ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),S)) ) ) ) ).

% pinf(10)
tff(fact_1835_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),S))
          <=> dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),S)) ) ) ) ).

% minf(9)
tff(fact_1836_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),S))
          <=> ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),S)) ) ) ) ).

% minf(10)
tff(fact_1837_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A3: A] :
          ( dvd_dvd(A,B2,A3)
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1838_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_1839_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_1840_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
          <=> dvd_dvd(A,B2,C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1841_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1842_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_1843_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_1844_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),one_one(A))
        <=> ( dvd_dvd(A,A3,one_one(A))
            & dvd_dvd(A,B2,one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_1845_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,D2: A,C2: A] :
          ( dvd_dvd(A,B2,A3)
         => ( dvd_dvd(A,D2,C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1846_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B2)
         => dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_1847_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A3: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_1848_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( dvd_dvd(A,C2,B2)
         => ( dvd_dvd(A,B2,A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_1849_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( dvd_dvd(A,C2,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ) ) ).

% div_mult_swap
tff(fact_1850_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( dvd_dvd(A,C2,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_1851_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1852_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),C2)
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1853_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_1854_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A3: A] :
          ( dvd_dvd(A,C2,B2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1855_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( dvd_dvd(A,C2,A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1856_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,Nb: nat] :
          ( dvd_dvd(A,B2,A3)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),Nb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% div_power
tff(fact_1857_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
         => dvd_dvd(A,B2,A3) ) ) ).

% mod_0_imp_dvd
tff(fact_1858_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,B2)
        <=> ( modulo_modulo(A,B2,A3) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1859_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
        <=> dvd_dvd(A,B2,A3) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1860_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_1861_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Nb: nat,B2: A,Mb: nat] :
          ( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb),B2)
         => ( aa(nat,$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),A3),Mb),B2) ) ) ) ).

% power_le_dvd
tff(fact_1862_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Mb: nat,Nb: nat,A3: A] :
          ( aa(nat,$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),A3),Mb),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ).

% le_imp_power_dvd
tff(fact_1863_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A3: A] : dvd_dvd(A,B2,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B2))) ) ).

% dvd_minus_mod
tff(fact_1864_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,B2,C2) )
        <=> dvd_dvd(A,C2,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ).

% mod_eq_dvd_iff
tff(fact_1865_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_1866_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_1867_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_1868_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_1869_dvd__diffD1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( dvd_dvd(nat,K,Mb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => dvd_dvd(nat,K,Nb) ) ) ) ).

% dvd_diffD1
tff(fact_1870_dvd__diffD,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( dvd_dvd(nat,K,Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => dvd_dvd(nat,K,Mb) ) ) ) ).

% dvd_diffD
tff(fact_1871_zdvd__not__zless,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Mb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Mb),Nb)
       => ~ dvd_dvd(int,Nb,Mb) ) ) ).

% zdvd_not_zless
tff(fact_1872_bezout__lemma__nat,axiom,
    ! [D2: nat,A3: nat,B2: nat,Xb: nat,Y: nat] :
      ( dvd_dvd(nat,D2,A3)
     => ( dvd_dvd(nat,D2,B2)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D2) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y)),D2) ) )
         => ? [X4: nat,Y4: nat] :
              ( dvd_dvd(nat,D2,A3)
              & dvd_dvd(nat,D2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),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),A3),B2)),Y4)),D2) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1873_bezout__add__nat,axiom,
    ! [A3: nat,B2: nat] :
    ? [D3: nat,X4: nat,Y4: nat] :
      ( dvd_dvd(nat,D3,A3)
      & dvd_dvd(nat,D3,B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D3) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_1874_bezout1__nat,axiom,
    ! [A3: nat,B2: nat] :
    ? [D3: nat,X4: nat,Y4: nat] :
      ( dvd_dvd(nat,D3,A3)
      & dvd_dvd(nat,D3,B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)) = D3 )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_1875_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) ) ) ).

% unit_dvdE
tff(fact_1876_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X))
        <=> ? [X: A] :
              ( dvd_dvd(A,L,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),zero_zero(A)))
              & aa(A,$o,P,X) ) ) ) ).

% unity_coeff_ex
tff(fact_1877_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,A3,B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1878_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( dvd_dvd(A,B2,A3)
           => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),C2)
            <=> dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1879_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( dvd_dvd(A,C2,B2)
           => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
            <=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1880_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( dvd_dvd(A,A3,B2)
             => ( dvd_dvd(A,C2,D2)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D2),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1881_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1882_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_1883_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D5: A,Ta: A] :
          ( dvd_dvd(A,D2,D5)
         => ! [X5: A,K4: A] :
              ( dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Ta))
            <=> dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),Ta)) ) ) ) ).

% inf_period(3)
tff(fact_1884_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D5: A,Ta: A] :
          ( dvd_dvd(A,D2,D5)
         => ! [X5: A,K4: A] :
              ( ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Ta))
            <=> ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),Ta)) ) ) ) ).

% inf_period(4)
tff(fact_1885_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,C2,one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_1886_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( dvd_dvd(A,C2,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_1887_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_1888_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( dvd_dvd(A,C2,one_one(A))
         => ( dvd_dvd(A,B2,A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_1889_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_1890_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = C2 )
          <=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_1891_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,Nb: nat] :
          ( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb),one_one(A))
        <=> ( dvd_dvd(A,A3,one_one(A))
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1892_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( modulo_modulo(A,A3,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1893_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( dvd_dvd(nat,K,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% dvd_imp_le
tff(fact_1894_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> dvd_dvd(nat,Mb,Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1895_dvd__mult__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => dvd_dvd(nat,Mb,Nb) ) ) ).

% dvd_mult_cancel
tff(fact_1896_bezout__add__strong__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [D3: nat,X4: nat,Y4: nat] :
          ( dvd_dvd(nat,D3,A3)
          & dvd_dvd(nat,D3,B2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1897_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( dvd_dvd(int,Z,Nb)
     => ( 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),Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_1898_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_1899_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( ( modulo_modulo(nat,Mb,Q2) = modulo_modulo(nat,Nb,Q2) )
      <=> dvd_dvd(nat,Q2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1900_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,Mb: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y5: A] :
              ( aa(A,$o,P,Y5)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Mb,X4)),aa(A,nat,Mb,Y5)) ) ) ) ).

% ex_has_least_nat
tff(fact_1901_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => finite_finite(nat,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ay(nat,fun(nat,$o),Mb))) ) ).

% finite_divisors_nat
tff(fact_1902_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_1903_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,B2,one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_1904_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,B2,one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_1905_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [B3: A] :
                  ( ( B3 != zero_zero(A) )
                 => ( dvd_dvd(A,B3,one_one(A))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = B3 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1906_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ~ ! [B3: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B3) ) ) ).

% evenE
tff(fact_1907_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_1908_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( ~ 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),A3),B2)) ) ) ) ).

% odd_even_add
tff(fact_1909_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
            <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) )
            & ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_1910_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_1911_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_1912_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_1913_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
     => ( ( 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_1914_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_1915_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_1916_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_1917_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_1918_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),Mb))
       => ( dvd_dvd(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Q2))
        <=> dvd_dvd(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),Mb)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_1919_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_1920_mod__nat__eqI,axiom,
    ! [R: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),Mb)
       => ( dvd_dvd(nat,Nb,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),R))
         => ( modulo_modulo(nat,Mb,Nb) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_1921_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L))
    <=> ( dvd_dvd(int,L,K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) )
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_1922_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = A3 ) ) ) ).

% even_two_times_div_two
tff(fact_1923_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1924_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1925_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A3: A,B2: A] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono_odd
tff(fact_1926_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_1927_dvd__power__iff__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_1928_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_1929_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A3: 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),A3))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
            | ( Mb = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_1930_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A3: 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),A3))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
            & ( Mb != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_1931_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),bit_se8732182000553998342ip_bit(A,Mb,A3))
        <=> ~ ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
            <=> ( Mb = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_1932_even__diff__iff,axiom,
    ! [K: 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),K),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),K),L)) ) ).

% even_diff_iff
tff(fact_1933_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ~ ! [B3: A] : A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B3)),one_one(A)) ) ) ).

% oddE
tff(fact_1934_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
           => ( modulo_modulo(A,A3,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)),A3)
             => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_1935_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          modulo_modulo(A,A3,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)),A3),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_1936_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A3: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ).

% zero_le_even_power
tff(fact_1937_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A3: A] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ) ).

% zero_le_odd_power
tff(fact_1938_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))
        <=> ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_le_power_eq
tff(fact_1939_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_1940_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_1941_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))
        <=> ( ( Nb = zero_zero(nat) )
            | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
              & ( A3 != zero_zero(A) ) )
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq
tff(fact_1942_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),B2: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y4: A] :
            ( aa(A,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y4)),B2) )
       => ? [X4: A] :
            ( aa(A,$o,P,X4)
            & ! [Y5: A] :
                ( aa(A,$o,P,Y5)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y5)),aa(A,nat,F2,X4)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_1943_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_1944_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% even_mask_div_iff'
tff(fact_1945_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),zero_zero(A))
        <=> ( aa(nat,$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),A3),zero_zero(A)) )
              | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1946_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,nat)] : aa(option(A),nat,size_option(A,Xb),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_1947_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% even_mod_4_div_2
tff(fact_1948_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_az(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_1949_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% even_mask_div_iff
tff(fact_1950_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ba(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_1951_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% odd_mod_4_div_2
tff(fact_1952_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)))
        <=> ( aa(nat,$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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_1953_divmod__step__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_bb(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_1954_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) ) )
           => ~ finite_finite(A,X6) ) ) ) ).

% infinite_growing
tff(fact_1955_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S3: set(A)] :
          ( finite_finite(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,S3)
                & ~ ? [Xa: A] :
                      ( member(A,Xa,S3)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X4) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1956_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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_1957_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_bc(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_1958_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_1959_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A3) = $ite(Nb = zero_zero(nat),aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,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,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))) ) ).

% signed_take_bit_rec
tff(fact_1960_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_1961_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% flip_bit_0
tff(fact_1962_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(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% set_decode_Suc
tff(fact_1963_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_1964_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A3)) = A3 ) ).

% add.inverse_inverse
tff(fact_1965_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A3 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_1966_case__prodI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A3: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F2,A3),B2)
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) ) ).

% case_prodI
tff(fact_1967_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A5: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A5),B3) )
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P2) ) ).

% case_prodI2
tff(fact_1968_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A3: B,B2: C] :
      ( member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C2,A3),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)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2))) ) ).

% mem_case_prodI
tff(fact_1969_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A5: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) )
         => member(C,Z,aa(B,set(C),aa(A,fun(B,set(C)),C2,A5),B3)) )
     => member(C,Z,aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P2)) ) ).

% mem_case_prodI2
tff(fact_1970_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),Xb: C] :
      ( ! [A5: A,B3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) = P2 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A5),B3),Xb) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P2),Xb) ) ).

% case_prodI2'
tff(fact_1971_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% neg_le_iff_le
tff(fact_1972_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_1973_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A3) )
        <=> ( zero_zero(A) = A3 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_1974_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_1975_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_1976_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = A3 )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_1977_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% neg_less_iff_less
tff(fact_1978_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_1979_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% mult_minus_left
tff(fact_1980_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) ) ).

% minus_mult_minus
tff(fact_1981_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% mult_minus_right
tff(fact_1982_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_1983_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_1984_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_1985_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).

% minus_diff_eq
tff(fact_1986_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ).

% div_minus_minus
tff(fact_1987_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_1988_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_1989_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,B2)) ) ).

% mod_minus_minus
tff(fact_1990_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_1991_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_1992_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_1993_real__add__minus__iff,axiom,
    ! [Xb: real,A3: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,uminus_uminus(real),A3)) = zero_zero(real) )
    <=> ( Xb = A3 ) ) ).

% real_add_minus_iff
tff(fact_1994_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_1995_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_1996_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_1997_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_1998_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_1999_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% neg_less_eq_nonneg
tff(fact_2000_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_2001_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% neg_le_0_iff_le
tff(fact_2002_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_2003_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_2004_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% neg_less_pos
tff(fact_2005_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_2006_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% neg_less_0_iff_less
tff(fact_2007_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),A3)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_2008_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_2009_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = aa(A,A,uminus_uminus(A),A3) ) ).

% diff_0
tff(fact_2010_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_2011_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_2012_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_2013_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) ) ).

% diff_minus_eq_add
tff(fact_2014_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).

% uminus_add_conv_diff
tff(fact_2015_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A3) ) ).

% div_minus1_right
tff(fact_2016_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Xb) ) ).

% divide_minus1
tff(fact_2017_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B2) ) ).

% minus_mod_self1
tff(fact_2018_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_2019_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_2020_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_2021_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_2022_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_2023_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2024_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,Nb) )
    <=> ( pred_numeral(K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_2025_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa(nat,nat,suc,Nb) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( Nb = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_2026_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_2027_dbl__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_simps(1)
tff(fact_2028_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_2029_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_2030_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_2031_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_2032_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_2033_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_2034_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_2035_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat,A3: 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)),A3)) = A3 ) ).

% left_minus_one_mult_self
tff(fact_2036_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : modulo_modulo(A,A3,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_2037_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_2038_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_2039_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_2040_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_2041_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_2042_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_2043_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_2044_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_2045_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_2046_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_2047_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_2048_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_2049_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_2050_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K)),Nb) ) ).

% less_numeral_Suc
tff(fact_2051_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit1,K)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_2052_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_2053_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),Nb) ) ).

% le_numeral_Suc
tff(fact_2054_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_2055_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K)),Nb) ).

% diff_numeral_Suc
tff(fact_2056_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_2057_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_2058_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_2059_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),Nb)) ).

% max_numeral_Suc
tff(fact_2060_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit0,K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_2061_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_2062_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_2063_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2064_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2065_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = A3 )
        <=> $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),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2066_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = B2,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2067_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2068_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2069_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_minus
tff(fact_2070_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_2071_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_2072_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_2073_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_2074_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% minus_1_div_2_eq
tff(fact_2075_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_2076_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_2077_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),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),A3),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_2078_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_2079_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A3: 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),A3)),Nb) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ) ) ).

% power_minus_odd
tff(fact_2080_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A3: 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),A3)),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_2081_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_2082_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_2083_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_2084_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_2085_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_2086_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_2087_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_2088_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% signed_take_bit_0
tff(fact_2089_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_2090_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_2091_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_2092_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_2093_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_2094_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_2095_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_2096_signed__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_2097_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,Mb,Nb)
     => ( dvd_dvd(nat,Nb,Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_2098_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o,Q2: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P2)) = aa($o,A,zero_neq_one_of_bool(A),(Q2)) )
        <=> ( (P2)
          <=> (Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_2099_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% equation_minus_iff
tff(fact_2100_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A3 ) ) ) ).

% minus_equation_iff
tff(fact_2101_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_2102_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z: A,C2: 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)),C2),P2))
     => ~ ! [X4: B,Y4: C] :
            ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y4) )
           => ~ member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C2,X4),Y4)) ) ) ).

% mem_case_prodE
tff(fact_2103_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_minus_iff
tff(fact_2104_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A3) ) ) ).

% minus_le_iff
tff(fact_2105_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_imp_neg_le
tff(fact_2106_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A3) ) ) ).

% minus_less_iff
tff(fact_2107_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% less_minus_iff
tff(fact_2108_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_2109_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_2110_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_2111_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
        <=> ( ( A3 = B2 )
            | ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% square_eq_iff
tff(fact_2112_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_mult_commute
tff(fact_2113_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_2114_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A4: A,K: A,A3: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,uminus_uminus(A),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).

% group_cancel.neg1
tff(fact_2115_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),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),A3)) ) ).

% add.inverse_distrib_swap
tff(fact_2116_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),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),A3)) ) ).

% is_num_normalize(8)
tff(fact_2117_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).

% minus_diff_commute
tff(fact_2118_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ).

% minus_diff_minus
tff(fact_2119_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).

% div_minus_right
tff(fact_2120_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).

% minus_divide_left
tff(fact_2121_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ).

% minus_divide_divide
tff(fact_2122_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_divide_right
tff(fact_2123_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,B2)),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B2) ) ).

% mod_minus_eq
tff(fact_2124_mod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A,A7: A] :
          ( ( modulo_modulo(A,A3,B2) = modulo_modulo(A,A7,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A7),B2) ) ) ) ).

% mod_minus_cong
tff(fact_2125_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,A3,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B2)) ) ).

% mod_minus_right
tff(fact_2126_case__prodD,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A3: A,B2: B] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))
     => aa(B,$o,aa(A,fun(B,$o),F2,A3),B2) ) ).

% case_prodD
tff(fact_2127_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P2: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P2)
     => ~ ! [X4: A,Y4: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X4),Y4) ) ) ).

% case_prodE
tff(fact_2128_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(A,fun(B,fun(C,$o))),A3: A,B2: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R2,A3),B2),C2) ) ).

% case_prodD'
tff(fact_2129_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,$o))),P2: product_prod(A,B),Z: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P2),Z)
     => ~ ! [X4: A,Y4: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X4),Y4),Z) ) ) ).

% case_prodE'
tff(fact_2130_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_2131_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_2132_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_2133_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_2134_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_2135_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_2136_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_2137_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_2138_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_2139_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_2140_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_2141_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_2142_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_2143_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% add_eq_0_iff
tff(fact_2144_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_2145_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A3) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_2146_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2147_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2148_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_2149_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_2150_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_2151_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_2152_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_2153_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2154_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_2155_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_2156_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B4: A,K: A,B2: A,A3: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_2157_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_2158_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2159_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_2160_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
     => ( ! [Y4: A] :
            ( member(A,Y4,aa(list(A),set(A),set2(A),Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate(A,Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_2161_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( dvd_dvd(A,B2,A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% dvd_div_neg
tff(fact_2162_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( dvd_dvd(A,B2,A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% dvd_neg_div
tff(fact_2163_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_2164_numeral__eq__Suc,axiom,
    ! [K: num] : aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,pred_numeral(K)) ).

% numeral_eq_Suc
tff(fact_2165_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_2166_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,K,aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_2167_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_2168_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_2169_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_2170_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_2171_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_2172_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_2173_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_2174_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_2175_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_2176_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_2177_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_2178_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_2179_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_2180_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_2181_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_2182_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_2183_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_2184_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_2185_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_2186_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2187_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2188_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A3 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_2189_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,uminus_uminus(A),B2),A3 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_2190_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2191_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_2192_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_2193_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_2194_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),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),A3),Nb)) ) ).

% power_minus
tff(fact_2195_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) ) ).

% power_minus_Bit0
tff(fact_2196_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_2197_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_2198_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_2199_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_2200_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_2201_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_2202_zmod__zminus2__eq__if,axiom,
    ! [A3: int,B2: int] :
      modulo_modulo(int,A3,aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A3,B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,A3,B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_2203_zmod__zminus1__eq__if,axiom,
    ! [A3: int,B2: int] :
      modulo_modulo(int,aa(int,int,uminus_uminus(int),A3),B2) = $ite(modulo_modulo(int,A3,B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),modulo_modulo(int,A3,B2))) ).

% zmod_zminus1_eq_if
tff(fact_2204_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
          <=> aa(A,$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),A3),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2205_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2206_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2207_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2208_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% minus_divide_less_eq
tff(fact_2209_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_2210_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2211_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2) = B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2212_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z))),B2) = $ite(Z = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2213_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2214_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_2215_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2216_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z)),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2217_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z))),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2218_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,uminus_uminus(A),A3))
        <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ).

% even_minus
tff(fact_2219_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_2220_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,Nb: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),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),A3),Nb),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb))) ) ).

% uminus_power_if
tff(fact_2221_verit__less__mono__div__int2,axiom,
    ! [A4: int,B4: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A4),B4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),Nb))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),Nb)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_2222_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_2223_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] : aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_2224_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
          <=> aa(A,$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),A3),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2225_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2226_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2227_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2228_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% minus_divide_le_eq
tff(fact_2229_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_2230_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2231_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2232_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
        <=> ( ( A3 = one_one(A) )
            | ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_2233_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_2234_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2235_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_2236_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2237_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2238_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2239_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),one_one(int))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2240_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_2241_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A3: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A3,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2242_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A3: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B2) = $ite(modulo_modulo(int,A3,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2243_zminus1__lemma,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B2,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R = zero_zero(int),aa(int,int,uminus_uminus(int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),
              $ite(R = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R)))) ) ) ).

% zminus1_lemma
tff(fact_2244_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A3: A] :
          ( ! [A5: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A5),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A5 )
             => aa(A,$o,P,A5) )
         => ( ! [A5: A,B3: $o] :
                ( aa(A,$o,P,A5)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A5))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A5 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A5))) ) )
           => aa(A,$o,P,A3) ) ) ) ).

% bits_induct
tff(fact_2245_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2246_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2247_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_2248_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: 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),A3)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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_2249_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2250_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2251_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_2252_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_2253_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_2254_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),modulo_modulo(nat,Mb,Nb)) ).

% divmod_nat_def
tff(fact_2255_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_2256_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K3: int] :
              ( aa(int,$o,P,K3)
             => ( ( K3 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) ) )
         => ( ! [K3: int] :
                ( aa(int,$o,P,K3)
               => ( ( K3 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_2257_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,Nb)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2258_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bd(nat,fun(nat,$o),Xb)) ).

% set_decode_def
tff(fact_2259_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb) != zero_zero(A) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))) ) ).

% exp_div_exp_eq
tff(fact_2260_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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_2261_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Nb))) ).

% one_div_minus_numeral
tff(fact_2262_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),Nb)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Nb))) ).

% minus_one_div_numeral
tff(fact_2263_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_2264_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_2265_numeral__div__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,Mb,Nb))) ).

% numeral_div_minus_numeral
tff(fact_2266_minus__numeral__div__numeral,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),aa(num,int,numeral_numeral(int),Nb)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,Mb,Nb))) ).

% minus_numeral_div_numeral
tff(fact_2267_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_2268_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_2269_dbl__dec__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% dbl_dec_simps(5)
tff(fact_2270_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_2271_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa($o,int,zero_neq_one_of_bool(int),R != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_2272_dbl__inc__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_inc_simps(1)
tff(fact_2273_dbl__dec__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_dec_simps(1)
tff(fact_2274_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),aTP_Lamp_be(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_2275_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_2276_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_2277_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_2278_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_2279_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_2280_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_2281_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_2282_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,uminus_uminus(real),Xb)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2283_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(
            K = zero_zero(int),
            zero_zero(A),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
              aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K))),
              $let(
                l: A,
                l:= aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),
                $ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = zero_zero(int),l,aa(A,A,aa(A,fun(A,A),plus_plus(A),l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_2284_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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va3))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_2285_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_bf(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_2286_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_bg(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_2287_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,numeral_numeral(int),K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_numeral
tff(fact_2288_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_2289_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_2290_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2291_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_2292_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_2293_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_2294_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_2295_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_2296_ln__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_2297_ln__inj__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( ( aa(real,real,ln_ln(real),Xb) = aa(real,real,ln_ln(real),Y) )
        <=> ( Xb = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_2298_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_2299_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_2300_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_2301_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_2302_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_2303_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_2304_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_2305_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_2306_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_2307_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_2308_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_2309_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_2310_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_2311_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_2312_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_2313_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_2314_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_2315_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_2316_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_2317_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_2318_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_2319_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_2320_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_2321_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_2322_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_2323_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_2324_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_2325_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$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)),A3) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2326_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),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_2327_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$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)),A3) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2328_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),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_2329_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_2330_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_2331_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),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_2332_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$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)),A3) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2333_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),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_2334_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$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)),A3) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2335_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_2336_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_2337_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_2338_ln__less__self,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),Xb) ) ).

% ln_less_self
tff(fact_2339_ln__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),Xb) ) ).

% ln_bound
tff(fact_2340_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_2341_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_2342_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_2343_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),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xb))),aa(real,real,aa(real,fun(real,real),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_2344_real__of__int__div,axiom,
    ! [D2: int,Nb: int] :
      ( dvd_dvd(int,D2,Nb)
     => ( aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),D2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Nb)),aa(int,real,ring_1_of_int(real),D2)) ) ) ).

% real_of_int_div
tff(fact_2345_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_2346_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_2347_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_2348_ln__div,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_2349_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_2350_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_2351_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_neg_numeral
tff(fact_2352_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_2353_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_2354_real__of__int__div__aux,axiom,
    ! [Xb: int,D2: int] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Xb)),aa(int,real,ring_1_of_int(real),D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),D2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),modulo_modulo(int,Xb,D2))),aa(int,real,ring_1_of_int(real),D2))) ).

% real_of_int_div_aux
tff(fact_2355_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_2356_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_2357_ln__diff__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_2358_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_2359_real__of__int__div2,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),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),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xb)))) ).

% real_of_int_div2
tff(fact_2360_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),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),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xb)))),one_one(real)) ).

% real_of_int_div3
tff(fact_2361_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_2362_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(int,A,ring_1_of_int(A),K))
        <=> dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K) ) ) ).

% even_of_int_iff
tff(fact_2363_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_2364_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A] : aa(A,A,artanh(A),Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Xb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% artanh_def
tff(fact_2365_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_2366_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,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),one_one(int))))
          & ! [Y5: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y5)),Xb)
                & aa(A,$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),Y5),one_one(int)))) )
             => ( Y5 = X4 ) ) ) ) ).

% floor_exists1
tff(fact_2367_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2368_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( tanh(real,aa(real,real,ln_ln(real),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2369_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2370_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_idempotent
tff(fact_2371_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_abs
tff(fact_2372_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2373_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_2374_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_2375_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_2376_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_2377_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% abs_mult_self_eq
tff(fact_2378_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_2379_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: 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),A3)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_2380_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_divide
tff(fact_2381_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_minus_cancel
tff(fact_2382_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_minus
tff(fact_2383_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: A,K: A] :
          ( dvd_dvd(A,Mb,aa(A,A,abs_abs(A),K))
        <=> dvd_dvd(A,Mb,K) ) ) ).

% dvd_abs_iff
tff(fact_2384_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: A,K: A] :
          ( dvd_dvd(A,aa(A,A,abs_abs(A),Mb),K)
        <=> dvd_dvd(A,Mb,K) ) ) ).

% abs_dvd_iff
tff(fact_2385_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_2386_tanh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),tanh(real,Xb)),tanh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% tanh_real_less_iff
tff(fact_2387_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_2388_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% abs_le_self_iff
tff(fact_2389_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2390_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2391_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_2392_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_2393_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: 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),A3)),Nb)) = aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ).

% abs_power_minus
tff(fact_2394_tanh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),tanh(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% tanh_real_pos_iff
tff(fact_2395_tanh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),tanh(real,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_2396_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2397_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2398_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_nonpos
tff(fact_2399_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_2400_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] :
          ( aa(A,$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),A3)),Nb))
        <=> ( ( A3 != zero_zero(A) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2401_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_abs
tff(fact_2402_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% abs_power2
tff(fact_2403_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A3: 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),A3)),aa(num,nat,numeral_numeral(nat),W)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W)) ) ) ) ).

% power_even_abs_numeral
tff(fact_2404_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_self
tff(fact_2405_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% abs_le_D1
tff(fact_2406_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_2407_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_mult
tff(fact_2408_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2409_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ).

% abs_minus_commute
tff(fact_2410_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_2411_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),Nb) ) ).

% power_abs
tff(fact_2412_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
         => dvd_dvd(A,L,K) ) ) ).

% dvd_if_abs_eq
tff(fact_2413_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_zero
tff(fact_2414_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_2415_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_2416_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$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),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq
tff(fact_2417_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ) ).

% abs_mult_less
tff(fact_2418_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$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),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))) ) ).

% abs_triangle_ineq2
tff(fact_2419_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$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),A3)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))) ) ).

% abs_triangle_ineq3
tff(fact_2420_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$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),A3)),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),A3))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2421_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_2422_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2) ) ) ) ).

% abs_leI
tff(fact_2423_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ) ).

% abs_le_D2
tff(fact_2424_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ) ) ).

% abs_le_iff
tff(fact_2425_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_minus_self
tff(fact_2426_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ) ) ).

% abs_less_iff
tff(fact_2427_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),tanh(real,Xb)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_2428_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_2429_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) )
            & ( aa(A,$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),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_2430_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_2431_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,abs_abs(A),B2) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
            & ( ( B2 = A3 )
              | ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2432_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A3) = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
            & ( ( A3 = B2 )
              | ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2433_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_2434_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),Xb)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% abs_div_pos
tff(fact_2435_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] : aa(A,$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),A3)),Nb)) ) ).

% zero_le_power_abs
tff(fact_2436_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X5: A] :
          aa(A,A,abs_abs(A),X5) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),zero_zero(A)),aa(A,A,uminus_uminus(A),X5),X5) ) ).

% abs_if_raw
tff(fact_2437_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: A] :
          aa(A,A,abs_abs(A),A3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)),aa(A,A,uminus_uminus(A),A3),A3) ) ).

% abs_if
tff(fact_2438_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_neg
tff(fact_2439_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A3: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A3))),R)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R)) ) ) ) ).

% abs_diff_le_iff
tff(fact_2440_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A,C2: A,D2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_2441_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$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),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq4
tff(fact_2442_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A3: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A3))),R)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R)) ) ) ) ).

% abs_diff_less_iff
tff(fact_2443_abs__real__def,axiom,
    ! [A3: real] :
      aa(real,real,abs_abs(real),A3) = $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),zero_zero(real)),aa(real,real,uminus_uminus(real),A3),A3) ).

% abs_real_def
tff(fact_2444_lemma__interval__lt,axiom,
    ! [A3: real,Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [Y5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y5))),D3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Y5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2445_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))),tanh(real,Xb)) ).

% tanh_real_gt_neg1
tff(fact_2446_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_2447_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_2448_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_2449_lemma__interval,axiom,
    ! [A3: real,Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [Y5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y5))),D3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Y5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_2450_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_2451_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_2452_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A3: 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),A3)),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb) ) ) ) ).

% power_even_abs
tff(fact_2453_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2454_abs__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_2455_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_2456_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_2457_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A3: A,B2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono_even
tff(fact_2458_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_2459_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_2460_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_2461_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_2462_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$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),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
           => ( archimedean_round(A,Xb) = Y ) ) ) ) ).

% round_unique
tff(fact_2463_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)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))
         => ( archimedean_round(A,Xb) = Nb ) ) ) ).

% round_unique'
tff(fact_2464_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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% of_int_round_abs_le
tff(fact_2465_arctan__double,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,arctan,Xb)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Xb)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% arctan_double
tff(fact_2466_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))) ) ).

% of_int_round_gt
tff(fact_2467_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))) ) ).

% of_int_round_ge
tff(fact_2468_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_2469_arctan__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_2470_zero__less__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arctan,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% zero_less_arctan_iff
tff(fact_2471_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_2472_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_2473_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_2474_arctan__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Xb)),aa(real,real,arctan,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% arctan_less_iff
tff(fact_2475_arctan__monotone,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Xb)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone
tff(fact_2476_abs__div,axiom,
    ! [Y: int,Xb: int] :
      ( dvd_dvd(int,Y,Xb)
     => ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),Xb)),aa(int,int,abs_abs(int),Y)) ) ) ).

% abs_div
tff(fact_2477_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_2478_zabs__def,axiom,
    ! [I: int] :
      aa(int,int,abs_abs(int),I) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_2479_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),modulo_modulo(int,K,L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_2480_even__add__abs__iff,axiom,
    ! [K: 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),K),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),K),L)) ) ).

% even_add_abs_iff
tff(fact_2481_even__abs__add__iff,axiom,
    ! [K: 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),K)),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),K),L)) ) ).

% even_abs_add_iff
tff(fact_2482_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_2483_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F2: fun(nat,int),K: 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,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,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,F2,Mb)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,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,F2,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2484_incr__lemma,axiom,
    ! [D2: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Z))),one_one(int))),D2))) ) ).

% incr_lemma
tff(fact_2485_decr__lemma,axiom,
    ! [D2: int,Xb: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Z))),one_one(int))),D2))),Z) ) ).

% decr_lemma
tff(fact_2486_nat__ivt__aux,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: 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,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2487_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: 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,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2488_arctan__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,arctan,Xb)),aa(real,real,arctan,Y)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_2489_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% of_int_round_le
tff(fact_2490_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R: A,A3: A,B2: A,C2: A,D2: A] :
          ( ( R != zero_zero(A) )
         => ( ( ( A3 = B2 )
              & ( C2 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2491_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_bh(int,int)),set_or1337092689740270186AtMost(int,Mb,Nb)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),Mb),aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),one_one(int))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ) ) ).

% Sum_Icc_int
tff(fact_2492_even__set__encode__iff,axiom,
    ! [A4: set(nat)] :
      ( finite_finite(nat,A4)
     => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(set(nat),nat,nat_set_encode,A4))
      <=> ~ member(nat,zero_zero(nat),A4) ) ) ).

% even_set_encode_iff
tff(fact_2493_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_2494_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_2495_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) = $ite(Nb = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),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,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% take_bit_rec
tff(fact_2496_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_2497_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_2498_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_concat_bit(Nb,K),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) ).

% concat_bit_of_zero_2
tff(fact_2499_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),A3) = zero_zero(A) ) ).

% take_bit_0
tff(fact_2500_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_2501_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_2502_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_2503_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_2504_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A4: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A4))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_bi(fun(B,A),fun(B,A),F2)),A4)) ) ).

% sum_abs
tff(fact_2505_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_2506_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_2507_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_2508_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_2509_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A4: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_bi(fun(B,A),fun(B,A),F2)),A4)) ) ).

% sum_abs_ge_zero
tff(fact_2510_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_2511_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3))
        <=> ( ( Nb = zero_zero(nat) )
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ) ).

% even_take_bit_eq
tff(fact_2512_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),A3) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_0
tff(fact_2513_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_2514_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_2515_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_of_int
tff(fact_2516_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_2517_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A3: 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),A3)),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),A3),B2)) ) ).

% take_bit_add
tff(fact_2518_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A,Mb: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_2519_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Mb),Q2)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_2520_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_2521_take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_2522_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% take_bit_mult
tff(fact_2523_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% take_bit_diff
tff(fact_2524_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_2525_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K: int,L: int,R: int,S: int] :
      ( ( aa(int,int,bit_concat_bit(Nb,K),L) = aa(int,int,bit_concat_bit(Nb,R),S) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),R) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_2526_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_2527_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K5: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,K5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),K5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),K5)) ) ) ).

% sum_mono
tff(fact_2528_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R: A,F2: fun(B,A),A4: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A4)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bj(A,fun(fun(B,A),fun(B,A)),R),F2)),A4) ) ).

% sum_distrib_left
tff(fact_2529_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A4: set(B),R: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A4)),R) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_bk(fun(B,A),fun(A,fun(B,A)),F2),R)),A4) ) ).

% sum_distrib_right
tff(fact_2530_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A4: set(B),G: fun(C,A),B4: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A4)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),B4)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_bm(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B4)),A4) ) ).

% sum_product
tff(fact_2531_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H: fun(B,A),A4: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bn(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A4)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),A4)) ) ).

% sum.distrib
tff(fact_2532_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A4: set(B),R: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A4)),R) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_bo(fun(B,A),fun(A,fun(B,A)),F2),R)),A4) ) ).

% sum_divide_distrib
tff(fact_2533_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A3: A,A4: set(B)] : modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_bp(fun(B,A),fun(A,fun(B,A)),F2),A3)),A4),A3) = modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A4),A3) ) ).

% mod_sum_eq
tff(fact_2534_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_2535_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_2536_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)) ) ) ).

% sum_nonneg
tff(fact_2537_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F2: fun(B,A),I5: set(B),G: fun(B,A),I: B] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),I5) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),I5) )
         => ( ! [I2: B] :
                ( member(B,I2,I5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,I2)),aa(B,A,G,I2)) )
           => ( member(B,I,I5)
             => ( finite_finite(B,I5)
               => ( aa(B,A,F2,I) = aa(B,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_2538_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2539_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A,B2: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A3) = aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),B2) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_2540_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_2541_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ).

% take_bit_nonnegative
tff(fact_2542_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_2543_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_2544_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)) = 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)),
            A3) ) ).

% signed_take_bit_take_bit
tff(fact_2545_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Mb),A3)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3))) ) ).

% take_bit_unset_bit_eq
tff(fact_2546_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Mb),A3)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3))) ) ).

% take_bit_set_bit_eq
tff(fact_2547_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se8732182000553998342ip_bit(A,Mb,A3)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3),bit_se8732182000553998342ip_bit(A,Mb,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3))) ) ).

% take_bit_flip_bit_eq
tff(fact_2548_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_2549_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_2550_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S: set(A),Ta: set(B),G: fun(B,C),I: fun(B,A),F2: fun(A,C)] :
          ( finite_finite(A,S)
         => ( finite_finite(B,Ta)
           => ( ! [X4: B] :
                  ( member(B,X4,Ta)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G,X4)) )
             => ( ! [X4: A] :
                    ( member(A,X4,S)
                   => ? [Xa: B] :
                        ( member(B,Xa,Ta)
                        & ( aa(B,A,I,Xa) = X4 )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X4)),aa(B,C,G,Xa)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,groups7311177749621191930dd_sum(A,C,F2),S)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_2551_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( ! [X4: A] :
                ( member(A,X4,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4) = zero_zero(B) )
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => ( aa(A,B,F2,X) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_2552_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( ! [X4: A] :
                ( member(A,X4,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
           => ( ? [X5: A] :
                  ( member(A,X5,A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X5)),aa(A,B,G,X5)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_2553_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R2: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,zero_zero(A)),zero_zero(A))
         => ( ! [X15: A,Y15: A,X2: A,Y22: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R2,X15),X2)
                  & aa(A,$o,aa(A,fun(A,$o),R2,Y15),Y22) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y22)) )
           => ( finite_finite(B,S3)
             => ( ! [X4: B] :
                    ( member(B,X4,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R2,aa(B,A,H,X4)),aa(B,A,G,X4)) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S3)) ) ) ) ) ) ).

% sum.related
tff(fact_2554_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X4: A] :
                  ( member(A,X4,A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4)) ) ) ) ) ).

% sum_strict_mono
tff(fact_2555_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] :
          ( aa(nat,$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),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),A3) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2556_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_2557_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),I: A] :
          ( finite_finite(A,S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S) = zero_zero(B) )
             => ( member(A,I,S)
               => ( aa(A,B,F2,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2558_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),B4: B,I: A] :
          ( finite_finite(A,S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S) = B4 )
             => ( member(A,I,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),B4) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2559_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_2560_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_2561_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( finite_finite(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_2562_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) ) ) ) ) ).

% sum_pos
tff(fact_2563_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B4: set(A),A4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A4)
         => ( finite_finite(A,A4)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B4)) ) ) ) ) ).

% sum.subset_diff
tff(fact_2564_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_bit0
tff(fact_2565_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) = modulo_modulo(A,A3,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_2566_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B4: set(A),A4: set(A),F2: fun(A,B)] :
          ( finite_finite(A,B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
           => ( ! [B3: A] :
                  ( member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A4))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,B3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B4)) ) ) ) ) ).

% sum_mono2
tff(fact_2567_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_2568_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_2569_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_2570_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_2571_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% take_bit_int_less_exp
tff(fact_2572_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = modulo_modulo(int,K,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% take_bit_int_def
tff(fact_2573_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_2574_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) = 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),A3) ) ) ).

% take_bit_eq_0_iff
tff(fact_2575_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B4: set(A),A4: set(A),B2: A,F2: fun(A,B)] :
          ( finite_finite(A,B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
           => ( member(A,B2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A4))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,B2))
               => ( ! [X4: A] :
                      ( member(A,X4,B4)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B4)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_2576_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_numeral_bit0
tff(fact_2577_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_2578_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_2579_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_2580_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_2581_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xb: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xb,I2)) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xb),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,I2)),B2))),Delta) )
             => aa(B,$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_bq(fun(A,B),fun(fun(A,B),fun(A,B)),Xb),A3)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2582_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2583_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_2584_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_2585_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_2586_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( ( ( A3 != B2 )
            & ( C2 != D2 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),D2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_2587_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_2588_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2589_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2590_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_2591_mask__half__int,axiom,
    ! [Nb: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,Nb)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ).

% mask_half_int
tff(fact_2592_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_2593_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),one_one(int)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_2594_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_2595_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_2596_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_2597_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_2598_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),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,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% take_bit_Suc
tff(fact_2599_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_2600_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_2601_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_2602_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_2603_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A3 )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) = $ite(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3),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_2604_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_2605_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_2606_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : size_num(aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_2607_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_2608_infinite__int__iff__unbounded,axiom,
    ! [S3: set(int)] :
      ( ~ finite_finite(int,S3)
    <=> ! [M5: int] :
        ? [N4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M5),aa(int,int,abs_abs(int),N4))
          & member(int,N4,S3) ) ) ).

% infinite_int_iff_unbounded
tff(fact_2609_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_2610_tanh__real__altdef,axiom,
    ! [Xb: real] : tanh(real,Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),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_2611_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) ) ),
        zero_zero(int),
        $ite(
          K = aa(int,int,uminus_uminus(int),one_one(int)),
          L,
          $ite(L = aa(int,int,uminus_uminus(int),one_one(int)),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% and_int_unfold
tff(fact_2612_power__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,L: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),K)),aa(num,nat,numeral_numeral(nat),L)) = aa(num,A,numeral_numeral(A),pow(K,L)) ) ).

% power_numeral
tff(fact_2613_and_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),A3) = A3 ) ).

% and.idem
tff(fact_2614_and_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2) ) ).

% and.left_idem
tff(fact_2615_and_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: 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),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2) ) ).

% and.right_idem
tff(fact_2616_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_2617_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_2618_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_2619_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_2620_exp__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),exp(real,Y)) ) ).

% exp_less_mono
tff(fact_2621_exp__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),exp(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% exp_less_cancel_iff
tff(fact_2622_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2)) ) ).

% take_bit_and
tff(fact_2623_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_2624_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_2625_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = A3 ) ).

% and.right_neutral
tff(fact_2626_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A3) = A3 ) ).

% and.left_neutral
tff(fact_2627_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_2628_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_2629_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_2630_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_2631_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_2632_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_2633_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_2634_exp__ln,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( exp(real,aa(real,real,ln_ln(real),Xb)) = Xb ) ) ).

% exp_ln
tff(fact_2635_exp__ln__iff,axiom,
    ! [Xb: real] :
      ( ( exp(real,aa(real,real,ln_ln(real),Xb)) = Xb )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% exp_ln_iff
tff(fact_2636_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_2637_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_2638_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_2639_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Mb),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ).

% sum.cl_ivl_Suc
tff(fact_2640_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_2641_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_2642_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_2643_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_2644_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_2645_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_2646_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A4: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_br(fun(nat,A),fun(nat,A),C2)),A4) = $ite(
            ( finite_finite(nat,A4)
            & member(nat,zero_zero(nat),A4) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_2647_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_2648_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_2649_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_2650_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_2651_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D2: fun(nat,A),A4: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A4) = $ite(
            ( finite_finite(nat,A4)
            & member(nat,zero_zero(nat),A4) ),
            aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D2,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_2652_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_2653_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_and_eq
tff(fact_2654_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.assoc
tff(fact_2655_and_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),A3) ) ).

% and.commute
tff(fact_2656_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.left_commute
tff(fact_2657_exp__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),exp(real,Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% exp_less_cancel
tff(fact_2658_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,A4)),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),exp(A,A4)) ) ).

% exp_times_arg_commute
tff(fact_2659_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_2660_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_2661_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A3 = 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_2662_not__exp__less__zero,axiom,
    ! [Xb: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_2663_exp__gt__zero,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),exp(real,Xb)) ).

% exp_gt_zero
tff(fact_2664_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X4: real] : exp(real,X4) = Y ) ).

% exp_total
tff(fact_2665_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_2666_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_2667_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_2668_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_2669_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_2670_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_2671_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_2672_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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),exp(A,Xb)),exp(A,Y)) ) ).

% exp_diff
tff(fact_2673_take__bit__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),bit_se2239418461657761734s_mask(A,Nb)) ) ).

% take_bit_eq_mask
tff(fact_2674_sum__subtractf__nat,axiom,
    ! [A: $tType,A4: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X4: A] :
          ( member(A,X4,A4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G,X4)),aa(A,nat,F2,X4)) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_bt(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A4) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A4)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_2675_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2676_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : pow(Xb,one2) = Xb ).

% pow.simps(1)
tff(fact_2677_inc_Osimps_I1_J,axiom,
    inc(one2) = aa(num,num,bit0,one2) ).

% inc.simps(1)
tff(fact_2678_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : inc(aa(num,num,bit1,Xb)) = aa(num,num,bit0,inc(Xb)) ).

% inc.simps(3)
tff(fact_2679_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : inc(aa(num,num,bit0,Xb)) = aa(num,num,bit1,Xb) ).

% inc.simps(2)
tff(fact_2680_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A4: set(A),Nb: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A4) = aa(nat,nat,suc,Nb) )
     => ? [X4: A] :
          ( member(A,X4,A4)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X4)) ) ) ).

% sum_SucD
tff(fact_2681_sum__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( finite_finite(A,A4)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A4) = one_one(nat) )
      <=> ? [X: A] :
            ( member(A,X,A4)
            & ( aa(A,nat,F2,X) = one_one(nat) )
            & ! [Xa2: A] :
                ( member(A,Xa2,A4)
               => ( ( X != Xa2 )
                 => ( aa(A,nat,F2,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2682_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_2683_add__One,axiom,
    ! [Xb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2) = inc(Xb) ).

% add_One
tff(fact_2684_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_2685_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_2686_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_2687_inc__BitM__eq,axiom,
    ! [Nb: num] : inc(bitM(Nb)) = aa(num,num,bit0,Nb) ).

% inc_BitM_eq
tff(fact_2688_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_2689_BitM__inc__eq,axiom,
    ! [Nb: num] : bitM(inc(Nb)) = aa(num,num,bit1,Nb) ).

% BitM_inc_eq
tff(fact_2690_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_2691_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,I5: set(nat)] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bv(A,fun(nat,fun(nat,A)),Xb),Mb)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),I5)) ) ).

% sum_power_add
tff(fact_2692_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Nb,Mb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_bw(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2693_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_bx(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_by(nat,fun(complex,$o),Nb))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_2694_sum__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_bx(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_ap(nat,fun(complex,fun(complex,$o)),Nb),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_2695_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: 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),A3),B2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_and_iff
tff(fact_2696_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2697_even__and__iff__int,axiom,
    ! [K: 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),K),L))
    <=> ( dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)
        | dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ) ) ).

% even_and_iff_int
tff(fact_2698_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2699_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2700_ln__ge__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_2701_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_2702_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bz(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2703_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ca(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,aa(nat,nat,suc,Nb))),aa(nat,A,F2,Mb)) ) ) ) ).

% sum_Suc_diff
tff(fact_2704_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),one_one(A)) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% and_one_eq
tff(fact_2705_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A3) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% one_and_eq
tff(fact_2706_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_2707_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2708_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_2709_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : tanh(A,Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),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_2710_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% exp_half_le2
tff(fact_2711_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_2712_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Mb)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_2713_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cc(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Mb)) ) ) ) ).

% sum_telescope''
tff(fact_2714_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,$o),aTP_Lamp_ak(nat,fun(nat,$o)),Nb))) ) ).

% mask_eq_sum_exp
tff(fact_2715_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_2716_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cd(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.in_pairs
tff(fact_2717_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set(nat)] :
      ( ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),M2)
         => ? [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N5)
              & member(nat,N5,S3) ) )
     => ~ finite_finite(nat,S3) ) ).

% unbounded_k_infinite
tff(fact_2718_infinite__nat__iff__unbounded,axiom,
    ! [S3: set(nat)] :
      ( ~ 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_2719_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set(nat)] :
      ( ~ 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_2720_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_2721_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_ak(nat,fun(nat,$o)),Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_2722_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)
            & ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% and_int_rec
tff(fact_2723_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ce(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% gauss_sum_nat
tff(fact_2724_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),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_2725_arith__series__nat,axiom,
    ! [A3: nat,D2: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_cf(nat,fun(nat,fun(nat,nat)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),D2)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% arith_series_nat
tff(fact_2726_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ce(nat,nat)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Icc_nat
tff(fact_2727_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),exp(real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_2728_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),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.elims
tff(fact_2729_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( member(int,K,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & member(int,L,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)
            & ~ 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)),K)
              & ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ))),
          aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ).

% and_int.simps
tff(fact_2730_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),
            zero_zero(A),
            $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))),Mb)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ) ).

% sum_gp
tff(fact_2731_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),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_2732_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,Nb))),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_2733_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_2734_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_2735_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_2736_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_2737_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_2738_height__double__log__univ__size,axiom,
    ! [U: real,Dega: nat,Ta: vEBT_VEBT] :
      ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Dega) )
     => ( vEBT_invar_vebt(Ta,Dega)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_2739_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_2740_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_2741_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_2742_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_2743_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_2744_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_2745_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_2746_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_2747_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_2748_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_2749_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_2750_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_2751_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_2752_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_2753_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_2754_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_2755_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_2756_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_2757_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_2758_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_2759_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( ~ member(A,Xb,A4)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,Xb),A4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4)) ) ) ) ) ).

% sum.insert
tff(fact_2760_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_2761_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_2762_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_2763_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_2764_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_2765_zero__less__log__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_2766_log__less__zero__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),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_2767_one__less__log__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_2768_log__less__one__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),Xb)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),A3) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_2769_log__less__cancel__iff,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),Xb)),aa(real,real,log(A3),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_2770_log__eq__one,axiom,
    ! [A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),A3) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_2771_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_2772_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),zero_zero(int))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_2773_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_2774_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_2775_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_2776_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_2777_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_2778_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_2779_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_2780_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_2781_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_2782_log__le__cancel__iff,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),Xb)),aa(real,real,log(A3),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_2783_log__le__one__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),Xb)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),A3) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_2784_one__le__log__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_2785_log__le__zero__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),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_2786_zero__le__log__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( 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(A3),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_2787_log__pow__cancel,axiom,
    ! [A3: real,B2: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_2788_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_2789_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_2790_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_2791_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_2792_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_2793_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),zero_zero(nat))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ).

% bit_0
tff(fact_2794_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_2795_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_2796_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_2797_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_2798_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_2799_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_2800_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_2801_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_2802_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_2803_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A3,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)),A3) ) ) ) ).

% bit_mod_2_iff
tff(fact_2804_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_2805_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_2806_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A,Nb: nat] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),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),A3),B2)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
              | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_2807_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_2808_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_2809_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_2810_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ).

% bit_and_iff
tff(fact_2811_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_2812_bit__and__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),Nb)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_2813_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Mb),A3)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
            & ( Mb != Nb ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_2814_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_2815_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_2816_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_2817_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_2818_log__base__pow,axiom,
    ! [A3: real,Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( aa(real,real,log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),Nb)),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A3),Xb)),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log_base_pow
tff(fact_2819_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_2820_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_2821_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_2822_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_2823_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_2824_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_2825_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),A3)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb) ) ) ) ).

% bit_take_bit_iff
tff(fact_2826_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_2827_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_2828_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,Mb: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),Mb))),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% div_mult2_eq'
tff(fact_2829_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_2830_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_2831_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_2832_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_2833_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_2834_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Nb: nat] :
          ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_2835_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_2836_nat__int__comparison_I2_J,axiom,
    ! [A3: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(2)
tff(fact_2837_nat__int__comparison_I3_J,axiom,
    ! [A3: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(3)
tff(fact_2838_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_2839_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_2840_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_2841_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_2842_int__ops_I5_J,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(5)
tff(fact_2843_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_2844_int__ops_I7_J,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_2845_not__int__zless__negative,axiom,
    ! [Nb: nat,Mb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Mb))) ).

% not_int_zless_negative
tff(fact_2846_zdiv__int,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zdiv_int
tff(fact_2847_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_2848_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_2849_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_2850_zmod__int,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,A3,B2)) = modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zmod_int
tff(fact_2851_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_2852_nat__less__as__int,axiom,
    ! [X5: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X5),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_less_as_int
tff(fact_2853_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_2854_nat__leq__as__int,axiom,
    ! [X5: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_leq_as_int
tff(fact_2855_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_2856_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_2857_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_2858_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int))),Nb)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% bit_not_int_iff'
tff(fact_2859_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S3: set(A),P: fun(set(A),$o),F2: fun(A,B)] :
          ( finite_finite(A,S3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X4: A,S4: set(A)] :
                  ( finite_finite(A,S4)
                 => ( ! [Y5: A] :
                        ( member(A,Y5,S4)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y5)),aa(A,B,F2,X4)) )
                   => ( aa(set(A),$o,P,S4)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),S4)) ) ) )
             => aa(set(A),$o,P,S3) ) ) ) ) ).

% finite_ranking_induct
tff(fact_2860_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),$o)] :
          ( finite_finite(A,A4)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B3: A,A8: set(A)] :
                  ( finite_finite(A,A8)
                 => ( ! [X5: A] :
                        ( member(A,X5,A8)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),X5) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B3),A8)) ) ) )
             => aa(set(A),$o,P,A4) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2861_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),$o)] :
          ( finite_finite(A,A4)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B3: A,A8: set(A)] :
                  ( finite_finite(A,A8)
                 => ( ! [X5: A] :
                        ( member(A,X5,A8)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),B3) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B3),A8)) ) ) )
             => aa(set(A),$o,P,A4) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2862_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_2863_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_2864_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,Xb),A4)) = $ite(member(A,Xb,A4),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4))) ) ) ) ).

% sum.insert_if
tff(fact_2865_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ! [Y5: real] :
        ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xb)) ) ).

% reals_Archimedean3
tff(fact_2866_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_2867_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xb))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xb))) ).

% real_of_nat_div4
tff(fact_2868_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_2869_real__of__nat__div,axiom,
    ! [D2: nat,Nb: nat] :
      ( dvd_dvd(nat,D2,Nb)
     => ( aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),D2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),D2)) ) ) ).

% real_of_nat_div
tff(fact_2870_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_2871_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_2872_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] :
          bit_se8732182000553998342ip_bit(A,Nb,A3) = aa(A,A,
            aa(nat,fun(A,A),
              $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),
              Nb),
            A3) ) ).

% flip_bit_eq_if
tff(fact_2873_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_2874_log__base__change,axiom,
    ! [A3: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(B2),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A3),Xb)),aa(real,real,log(A3),B2)) ) ) ) ).

% log_base_change
tff(fact_2875_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,Mb: nat,Nb: nat] : modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)))),modulo_modulo(A,A3,aa(nat,A,semiring_1_of_nat(A),Mb))) ) ).

% mod_mult2_eq'
tff(fact_2876_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_2877_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,Mb,Nb)))),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% field_char_0_class.of_nat_div
tff(fact_2878_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_concat_bit(Mb,K),L)),Nb)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_2879_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ~ ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% pos_int_cases
tff(fact_2880_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2881_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) )
       => ~ ! [N: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% int_cases3
tff(fact_2882_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_2883_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_2884_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: 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)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_2885_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_2886_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_2887_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_2888_finite__induct__select,axiom,
    ! [A: $tType,S3: set(A),P: fun(set(A),$o)] :
      ( finite_finite(A,S3)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T2: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T2),S3)
             => ( aa(set(A),$o,P,T2)
               => ? [X5: A] :
                    ( member(A,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T2))
                    & aa(set(A),$o,P,aa(set(A),set(A),insert(A,X5),T2)) ) ) )
         => aa(set(A),$o,P,S3) ) ) ) ).

% finite_induct_select
tff(fact_2889_psubset__insert__iff,axiom,
    ! [A: $tType,A4: set(A),Xb: A,B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),aa(set(A),set(A),insert(A,Xb),B4))
    <=> $ite(
          member(A,Xb,B4),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4),
          $ite(member(A,Xb,A4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_2890_int__ops_I6_J,axiom,
    ! [A3: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),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),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) ).

% int_ops(6)
tff(fact_2891_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D2: nat] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Xb)),aa(nat,real,semiring_1_of_nat(real),D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),D2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),modulo_modulo(nat,Xb,D2))),aa(nat,real,semiring_1_of_nat(real),D2))) ).

% real_of_nat_div_aux
tff(fact_2892_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_2893_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = aa(int,int,bit_concat_bit(Nb,K),aa(int,int,uminus_uminus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_2894_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) = zero_zero(A) )
         => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_2895_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Nb) ) ) ).

% bit_Suc
tff(fact_2896_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A3 )
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
          <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_2897_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
            <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A3 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_2898_log__mult,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != 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(A3),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(A3),Xb)),aa(real,real,log(A3),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_2899_log__divide,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != 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(A3),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(A3),Xb)),aa(real,real,log(A3),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_2900_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),M)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_2901_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
         => ~ ! [N: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),E) ) ) ).

% nat_approx_posE
tff(fact_2902_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_2903_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: nat,Mb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => ( ( Nb != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Mb))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ) ) ).

% inverse_of_nat_le
tff(fact_2904_insert__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(Ta,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_2905_member__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(Ta,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_2906_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(nat,A,semiring_1_of_nat(A),Nb)))),Nb) = exp(A,Xb) ) ) ) ).

% exp_divide_power_eq
tff(fact_2907_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
       => ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
             => 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),M2)),Xb)),C2) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2908_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,Xb),A4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_2909_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( member(A,Xb,A4)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_2910_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ~ ! [N: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% neg_int_cases
tff(fact_2911_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_2912_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xb))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xb)))) ).

% real_of_nat_div2
tff(fact_2913_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xb))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xb)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_2914_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_2915_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),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_2916_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A3: A,B2: fun(A,B),C2: fun(A,B)] :
          ( 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_cg(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B2),C2)),S3) = $ite(member(A,A3,S3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_2917_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = zero_zero(A) )
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_2918_log__eq__div__ln__mult__log,axiom,
    ! [A3: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != 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(A3),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(real,real,ln_ln(real),A3))),aa(real,real,log(B2),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2919_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
    <=> ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))) ) ).

% bit_int_def
tff(fact_2920_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A4: set(A),F2: fun(A,B)] :
          ( member(A,I,A4)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
           => ( finite_finite(A,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)) ) ) ) ) ).

% member_le_sum
tff(fact_2921_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_2922_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_2923_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_2924_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_2925_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,Nb: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),Nb)
          <=> $ite(Nb = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3),aa(nat,$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_2926_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
        <=> $ite(Nb = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),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_2927_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Mb),Nb) = $ite(
        ( ( Mb = zero_zero(nat) )
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% and_nat_unfold
tff(fact_2928_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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% and_nat_rec
tff(fact_2929_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,D2: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ch(A,fun(A,fun(nat,A)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),D2))) ) ).

% double_arith_series
tff(fact_2930_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_2931_set__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))) ).

% set_bit_eq
tff(fact_2932_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))) ).

% unset_bit_eq
tff(fact_2933_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_2934_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_2935_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,D2: 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)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),D2)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% arith_series
tff(fact_2936_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum
tff(fact_2937_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) = $ite(Xb = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ).

% sum_gp_offset
tff(fact_2938_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_2939_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Nb))),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_2940_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,aa(real,real,uminus_uminus(real),Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_2941_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_2942_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_2943_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_2944_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K5: real,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K5)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2945_log__ceil__idem,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Xb)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(int,real,ring_1_of_int(real),archimedean_ceiling(real,Xb)))) ) ) ).

% log_ceil_idem
tff(fact_2946_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_cl(real,fun(nat,real),Xb)) ) ) ) ).

% ln_series
tff(fact_2947_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cn(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_2948_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_lessThan(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),K) ) ) ).

% lessThan_iff
tff(fact_2949_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_2950_heigt__uplog__rel,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ).

% heigt_uplog_rel
tff(fact_2951_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_2952_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_2953_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_2954_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,Nb))),aa(nat,A,G,Nb)) ) ).

% sum.lessThan_Suc
tff(fact_2955_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_2956_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_2957_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_2958_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_2959_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_2960_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_2961_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_2962_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_2963_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_2964_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_2965_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_2966_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_2967_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_2968_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_2969_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_2970_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_co(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2971_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_2972_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_2973_ceiling__divide__eq__div__numeral,axiom,
    ! [A3: num,B2: num] : archimedean_ceiling(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_divide_eq_div_numeral
tff(fact_2974_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_2975_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_2976_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_2977_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_2978_set__encode__insert,axiom,
    ! [A4: set(nat),Nb: nat] :
      ( finite_finite(nat,A4)
     => ( ~ member(nat,Nb,A4)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,Nb),A4)) = 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,A4)) ) ) ) ).

% set_encode_insert
tff(fact_2979_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A3: num,B2: num] : archimedean_ceiling(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_2980_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_2981_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_2982_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_ord_lessThan(nat,pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_2983_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cp(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_2984_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_2985_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_2986_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_2987_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_2988_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_2989_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_2990_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_2991_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A3))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),A3) ) ) ).

% ceiling_le
tff(fact_2992_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_2993_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_2994_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_2995_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_2996_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_2997_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_2998_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_cq(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,Nb)) ) ) ) ).

% sum_diff_distrib
tff(fact_2999_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_3000_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R))),one_one(A))),R) ) ).

% of_int_ceiling_diff_one_le
tff(fact_3001_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,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),K6) )
        <=> ? [N6: nat] :
            ! [N4: A] : aa(real,$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,N6))) ) ) ).

% lemma_NBseq_def
tff(fact_3002_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,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),K6) )
        <=> ? [N6: nat] :
            ! [N4: A] : aa(real,$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,N6))) ) ) ).

% lemma_NBseq_def2
tff(fact_3003_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: int,B2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B2)) ) ).

% ceiling_divide_eq_div
tff(fact_3004_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bz(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3005_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),Nb: nat,R: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),R)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),F2),R)),set_ord_lessThan(nat,Nb)) ) ).

% sumr_diff_mult_const2
tff(fact_3006_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),aa(int,A,ring_1_of_int(A),I4)) )
             => aa(int,$o,P,I4) ) ) ) ).

% ceiling_split
tff(fact_3007_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A3: int] :
          ( ( archimedean_ceiling(A,Xb) = A3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A3)) ) ) ) ).

% ceiling_eq_iff
tff(fact_3008_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_3009_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_3010_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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),A3),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B2))) ) ) ) ).

% mult_ceiling_le
tff(fact_3011_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_3012_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_3013_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_3014_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_3015_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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3016_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),Q2)) ) ) ).

% ceiling_divide_upper
tff(fact_3017_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)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))) ) ).

% bit_nat_def
tff(fact_3018_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_3019_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ).

% sum_gp_strict
tff(fact_3020_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_cs(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_ct(A,fun(A,fun(nat,fun(nat,A))),Z),H),Mb)),set_ord_lessThan(nat,Mb)) ) ).

% lemma_termdiff1
tff(fact_3021_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_cu(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),set_ord_lessThan(nat,Nb))) ) ).

% power_diff_sumr2
tff(fact_3022_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_cv(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_3023_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_3024_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),one_one(A))),Q2)),P2) ) ) ).

% ceiling_divide_lower
tff(fact_3025_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_3026_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,F2: fun(nat,A),K5: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,P4)),K5) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K5)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_3027_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% exp_bound_half
tff(fact_3028_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_cw(A,fun(nat,fun(nat,A)),Xb),Nb)),set_ord_lessThan(nat,Nb))) ) ).

% one_diff_power_eq'
tff(fact_3029_sum__split__even__odd,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real),Nb: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_cx(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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_cy(fun(nat,real),fun(nat,real),F2)),set_ord_lessThan(nat,Nb))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_cz(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,Nb))) ).

% sum_split_even_odd
tff(fact_3030_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,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_3031_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_3032_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_3033_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_3034_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_da(real,fun(nat,real),Xb)) ) ) ).

% arctan_series
tff(fact_3035_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A3)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_3036_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W: num,A3: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W)),real_V7770717601297561774m_norm(A,A3)) ) ).

% norm_mult_numeral1
tff(fact_3037_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A3)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_mult_numeral2
tff(fact_3038_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_3039_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xb))
        <=> ( Xb != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_3040_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_3041_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_3042_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_3043_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_3044_norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A,B2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B2)) ) ).

% norm_divide
tff(fact_3045_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_3046_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_3047_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_3048_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_3049_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A,R: real,Y: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R),S)) ) ) ) ).

% norm_mult_less
tff(fact_3050_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_3051_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,R: real,Y: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R),S)) ) ) ) ).

% norm_add_less
tff(fact_3052_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E) ) ) ).

% norm_triangle_lt
tff(fact_3053_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A,C2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),C2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A3)),C2)) ) ) ).

% norm_add_leD
tff(fact_3054_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E) ) ) ).

% norm_triangle_le
tff(fact_3055_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_3056_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,R: real,B2: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A3)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),S)
           => aa(real,$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),A3),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R),S)) ) ) ) ).

% norm_triangle_mono
tff(fact_3057_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_3058_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_3059_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A] : aa(real,$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,A3)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ).

% norm_diff_ineq
tff(fact_3060_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_3061_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A,C2: A,D2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% norm_diff_triangle_ineq
tff(fact_3062_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_3063_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_3064_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => ( suminf(A,aa(A,fun(nat,A),power_power(A),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_3065_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mm: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bz(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_3066_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_db(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_3067_pi__series,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = suminf(real,aTP_Lamp_dc(nat,real)) ).

% pi_series
tff(fact_3068_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_da(real,fun(nat,real),Xb)) ) ).

% summable_arctan_series
tff(fact_3069_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_3070_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_de(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_3071_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_3072_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,aa(A,fun(nat,A),power_power(A),C2))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_3073_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_3074_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_3075_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_dh(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult
tff(fact_3076_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_di(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult2
tff(fact_3077_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_3078_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test
tff(fact_3079_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N2: nat,F2: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
           => summable(A,F2) ) ) ) ).

% summable_comparison_test'
tff(fact_3080_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,G,N))
         => ( summable(A,F2)
           => ( summable(A,G)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),suminf(A,G)) ) ) ) ) ).

% suminf_le
tff(fact_3081_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_de(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_3082_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_3083_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F2)),C2) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_di(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ) ).

% suminf_mult2
tff(fact_3084_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_dh(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),suminf(A,F2)) ) ) ) ).

% suminf_mult
tff(fact_3085_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_3086_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),suminf(A,F2)),C2) ) ) ) ).

% suminf_divide
tff(fact_3087_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_dj(fun(nat,A),fun(A,fun(nat,A)),F2),Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_dk(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_3088_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_nonneg
tff(fact_3089_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N4: nat] : aa(nat,A,F2,N4) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3090_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_pos
tff(fact_3091_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_dl(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_3092_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_dm(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_3093_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_dn(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_dj(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_3094_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_dp(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% powser_split_head(3)
tff(fact_3095_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Mb: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dq(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),Mb),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_dj(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_3096_pi__not__less__zero,axiom,
    ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_3097_pi__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_3098_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_dr(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3099_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F2,N))),aa(nat,real,G,N)) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_ds(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_3100_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2))
            <=> ? [I4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I4)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3101_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I: nat] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ) ).

% suminf_pos2
tff(fact_3102_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xb: A] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),Xb) ) ) ) ).

% suminf_le_const
tff(fact_3103_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xb: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))),Xb)
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3104_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_3105_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),C2)) ) ) ).

% summable_geometric
tff(fact_3106_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I5: set(nat)] :
          ( summable(A,F2)
         => ( finite_finite(nat,I5)
           => ( ! [N: nat] :
                  ( member(nat,N,aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),I5)),suminf(A,F2)) ) ) ) ) ).

% sum_le_suminf
tff(fact_3107_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,F2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,A)),F2),K))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_3108_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_3109_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(A,fun(nat,A)),F2),Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_3110_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat] :
          ( summable(A,F2)
         => ( ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,M2)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb))),suminf(A,F2)) ) ) ) ).

% sum_less_suminf
tff(fact_3111_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_3112_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_3113_pi__half__neq__two,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)) ).

% pi_half_neq_two
tff(fact_3114_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F2,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_dp(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3115_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_dp(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3116_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E: real] :
          ( summable(A,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => ~ ! [N8: nat] :
                  ~ ! [M: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),M)
                     => ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,M,N5)))),E) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_3117_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => ( summable(A,F2)
           => ? [N8: nat] :
              ! [N5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,A)),F2),N5)))),R) ) ) ) ) ).

% suminf_exist_split
tff(fact_3118_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,I2)),one_one(real))
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,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_dt(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3119_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,R0: real,A3: fun(nat,A),M6: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),R),R0)
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A3,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_du(real,fun(fun(nat,A),fun(nat,real)),R),A3)) ) ) ) ) ).

% Abel_lemma
tff(fact_3120_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N2: nat,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,aa(nat,nat,suc,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N)))) )
           => summable(A,F2) ) ) ) ).

% summable_ratio_test
tff(fact_3121_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M2)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),I)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb))),suminf(A,F2)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3122_pi__half__neq__zero,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_3123_pi__half__less__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% pi_half_less_two
tff(fact_3124_pi__half__le__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% pi_half_le_two
tff(fact_3125_pi__half__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% pi_half_gt_zero
tff(fact_3126_pi__half__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% pi_half_ge_zero
tff(fact_3127_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_3128_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% arctan_ubound
tff(fact_3129_arctan__one,axiom,
    aa(real,real,arctan,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) ).

% arctan_one
tff(fact_3130_minus__pi__half__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_3131_arctan__lbound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y)) ).

% arctan_lbound
tff(fact_3132_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ).

% arctan_bounded
tff(fact_3133_machin__Euler,axiom,
    aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) ).

% machin_Euler
tff(fact_3134_machin,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit0,one2))))))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))) ).

% machin
tff(fact_3135_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)))),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,F2),set_ord_lessThan(nat,K))),suminf(real,F2)) ) ) ).

% sum_pos_lt_pair
tff(fact_3136_sin__cos__npi,axiom,
    ! [Nb: nat] : sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Nb) ).

% sin_cos_npi
tff(fact_3137_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb))))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_3138_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),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_3139_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
            ( member(int,K,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,L,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)
                & ~ 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)),K)
                  & ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_3140_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)
       => ? [T3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T3))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T3)),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_dv(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),exp(real,T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3141_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_3142_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3143_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_3144_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_3145_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_3146_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_3147_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_3148_cos__pi__half,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_3149_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_3150_sin__pi__half,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_3151_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_3152_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_3153_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_3154_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_3155_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_3156_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_3157_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_3158_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_3159_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_3160_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_3161_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_3162_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_3163_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_3164_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_3165_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% sin_3over2_pi
tff(fact_3166_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_3167_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_3168_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_3169_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_3170_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_3171_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_3172_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_3173_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y4)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),pi)
      & ( sin(real,Y4) = sin(real,Xb) )
      & ( cos(real,Y4) = cos(real,Xb) ) ) ).

% sincos_principal_value
tff(fact_3174_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_3175_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_3176_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_3177_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_3178_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_3179_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_3180_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_3181_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_3182_sin__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero
tff(fact_3183_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_3184_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sin_times_sin
tff(fact_3185_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sin_times_cos
tff(fact_3186_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% cos_times_sin
tff(fact_3187_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% sin_plus_sin
tff(fact_3188_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% sin_diff_sin
tff(fact_3189_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% cos_diff_cos
tff(fact_3190_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] : dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))) ) ).

% fact_fact_dvd_fact
tff(fact_3191_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_3192_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_3193_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_3194_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_3195_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_3196_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xb)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3197_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xb)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3198_sin__eq__0__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),pi)
       => ( ( sin(real,Xb) = zero_zero(real) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3199_sin__zero__pi__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),pi)
     => ( ( sin(real,Xb) = zero_zero(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3200_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))),semiring_char_0_fact(A,Nb)) ) ) ).

% choose_dvd
tff(fact_3201_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_3202_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) )
       => ? [T3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),pi)
            & ( Xb = cos(real,T3) )
            & ( Y = sin(real,T3) ) ) ) ) ).

% sincos_total_pi
tff(fact_3203_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Mb))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Mb)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% sin_expansion_lemma
tff(fact_3204_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Mb))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = aa(real,real,uminus_uminus(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Mb)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% cos_expansion_lemma
tff(fact_3205_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_3206_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_3207_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) )
      & ! [Y5: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
            & ( cos(real,Y5) = zero_zero(real) ) )
         => ( Y5 = X4 ) ) ) ).

% cos_is_zero
tff(fact_3208_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_3209_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Y)),cos(real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3210_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) )
         => ? [T3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T3)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
              & ( Xb = cos(real,T3) )
              & ( Y = sin(real,T3) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_3211_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) )
     => ? [T3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),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,T3) )
          & ( Y = sin(real,T3) ) ) ) ).

% sincos_total_2pi_le
tff(fact_3212_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_3213_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) )
     => ~ ! [T3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),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,T3) )
               => ( Y != sin(real,T3) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3214_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_3215_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_3216_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% cos_times_cos
tff(fact_3217_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% cos_plus_cos
tff(fact_3218_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_3219_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_3220_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_3221_sin__30,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_30
tff(fact_3222_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_3223_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3224_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3225_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( ( sin(real,Xb) = sin(real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3226_cos__60,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_60
tff(fact_3227_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = one_one(real) )
    <=> ? [X: int] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3228_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_3229_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_3230_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),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_dw(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_3231_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_3232_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_3233_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3234_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3235_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),aa(real,real,aa(real,fun(real,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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
            & ( sin(real,X4) = Y )
            & ! [Y5: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
                  & ( sin(real,Y5) = Y ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% sin_total
tff(fact_3236_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_3237_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_3238_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = one_one(real) )
    <=> ( ? [X: nat] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)
        | ? [X: nat] : Xb = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_3239_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K3: int,L3: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K3),L3))
           => ( ( ~ ( member(int,K3,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L3,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K3),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L3),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K3),L3) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_3240_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_dx(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_3241_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ? [B9: real] : aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_dy(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),B9),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb)),semiring_char_0_fact(real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_3242_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))
       => ? [T3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),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_dw(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3243_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)
       => ? [T3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),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_dw(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_3244_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_3245_Maclaurin__exp__le,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),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_dv(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),exp(real,T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_3246_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I4: int] :
          ( dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),I4)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_3247_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = zero_zero(real) )
    <=> ? [I4: int] :
          ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),I4)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_3248_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)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_3249_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)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N4: nat] :
            ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N4)
            & ( Xb = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_3250_cos__coeff__def,axiom,
    ! [X5: nat] :
      cos_coeff(X5) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X5),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X5),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),semiring_char_0_fact(real,X5)),zero_zero(real)) ).

% cos_coeff_def
tff(fact_3251_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)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_3252_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)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N4: nat] :
            ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N4)
            & ( Xb = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_3253_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)
       => ? [T3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),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_dz(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_3254_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ? [T3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),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_dz(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_3255_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),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_dz(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_3256_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: 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_dz(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_3257_sin__coeff__def,axiom,
    ! [X5: nat] :
      sin_coeff(X5) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X5),zero_zero(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X5),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),semiring_char_0_fact(real,X5))) ).

% sin_coeff_def
tff(fact_3258_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_3259_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_3260_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_3261_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_3262_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_3263_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_3264_fact__div__fact__le__pow,axiom,
    ! [R: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Nb)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),R)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R)) ) ).

% fact_div_fact_le_pow
tff(fact_3265_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,tan(A),Xb))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ) ).

% tan_double
tff(fact_3266_sin__paired,axiom,
    ! [Xb: real] : sums(real,aTP_Lamp_ea(real,fun(nat,real),Xb),sin(real,Xb)) ).

% sin_paired
tff(fact_3267_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),Xb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),Xb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_3268_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_eb(A,fun(nat,A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_3269_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M2: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,M2)),aa(nat,A,X6,N)) )
         => topological_monoseq(A,X6) ) ) ).

% monoI1
tff(fact_3270_powr__one__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [A3: A] : powr(A,one_one(A),A3) = one_one(A) ) ).

% powr_one_eq_one
tff(fact_3271_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_3272_powr__gt__zero,axiom,
    ! [Xb: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),powr(real,Xb,A3))
    <=> ( Xb != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_3273_powr__less__cancel__iff,axiom,
    ! [Xb: real,A3: 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,A3)),powr(real,Xb,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_3274_powr__eq__one__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( ( powr(real,A3,Xb) = one_one(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_3275_powr__le__cancel__iff,axiom,
    ! [Xb: real,A3: 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,A3)),powr(real,Xb,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_3276_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_3277_powr__log__cancel,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( powr(real,A3,aa(real,real,log(A3),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_3278_log__powr__cancel,axiom,
    ! [A3: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),powr(real,A3,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_3279_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A),Xb: A] :
          ( sums(A,aTP_Lamp_dm(fun(nat,A),fun(nat,A),A3),Xb)
        <=> ( aa(nat,A,A3,zero_zero(nat)) = Xb ) ) ) ).

% powser_sums_zero_iff
tff(fact_3280_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_3281_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_3282_square__powr__half,axiom,
    ! [Xb: real] : powr(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(real,real,abs_abs(real),Xb) ).

% square_powr_half
tff(fact_3283_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A),S: A,Ta: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,G,N))
         => ( sums(A,F2,S)
           => ( sums(A,G,Ta)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S),Ta) ) ) ) ) ).

% sums_le
tff(fact_3284_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A3: A,C2: A] :
          ( sums(A,F2,A3)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_di(fun(nat,A),fun(A,fun(nat,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ).

% sums_mult2
tff(fact_3285_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A3: A,C2: A] :
          ( sums(A,F2,A3)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_dh(fun(nat,A),fun(A,fun(nat,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) ) ) ).

% sums_mult
tff(fact_3286_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),A3: A,G: fun(nat,A),B2: A] :
          ( sums(A,F2,A3)
         => ( sums(A,G,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% sums_add
tff(fact_3287_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),A3: A,C2: A] :
          ( sums(A,F2,A3)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)) ) ) ).

% sums_divide
tff(fact_3288_powr__non__neg,axiom,
    ! [A3: real,Xb: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,A3,Xb)),zero_zero(real)) ).

% powr_non_neg
tff(fact_3289_powr__less__mono2__neg,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Y,A3)),powr(real,Xb,A3)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_3290_powr__less__mono,axiom,
    ! [A3: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),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,A3)),powr(real,Xb,B2)) ) ) ).

% powr_less_mono
tff(fact_3291_powr__less__cancel,axiom,
    ! [Xb: real,A3: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A3)),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),A3),B2) ) ) ).

% powr_less_cancel
tff(fact_3292_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ec(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),C2))
          <=> sums(A,F2,D2) ) ) ) ).

% sums_mult2_iff
tff(fact_3293_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ed(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))
          <=> sums(A,F2,D2) ) ) ) ).

% sums_mult_iff
tff(fact_3294_powr__less__mono2,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_less_mono2
tff(fact_3295_powr__mono2_H,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Y,A3)),powr(real,Xb,A3)) ) ) ) ).

% powr_mono2'
tff(fact_3296_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_3297_powr__inj,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( ( powr(real,A3,Xb) = powr(real,A3,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_3298_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A3: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_de(A,fun(fun(nat,A),fun(nat,A)),C2),F2),A3)
         => ( ( C2 != zero_zero(A) )
           => sums(A,F2,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)) ) ) ) ).

% sums_mult_D
tff(fact_3299_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,aTP_Lamp_ee(fun(nat,A),fun(nat,A),F2),S)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_3300_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_ef(fun(nat,A),fun(nat,A),F2),L)
         => sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_3301_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Nb: nat,F2: fun(nat,A),S: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
             => ( aa(nat,A,F2,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),F2),S)
          <=> sums(A,F2,S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3302_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A3: A,B2: A] : powr(A,Xb,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),powr(A,Xb,A3)),powr(A,Xb,B2)) ) ).

% powr_add
tff(fact_3303_powr__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [W: A,Z1: A,Z22: A] : powr(A,W,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z1),Z22)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),powr(A,W,Z1)),powr(A,W,Z22)) ) ).

% powr_diff
tff(fact_3304_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_3305_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A)] : sums(A,aTP_Lamp_dm(fun(nat,A),fun(nat,A),A3),aa(nat,A,A3,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3306_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X5: A] : aa(A,A,tan(A),X5) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,X5)),cos(A,X5)) ) ).

% tan_def
tff(fact_3307_powr__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb) ) ) ).

% powr_realpow
tff(fact_3308_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_3309_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_3310_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_3311_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_3312_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),S)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_iff_shift
tff(fact_3313_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb))))
        <=> sums(A,F2,S) ) ) ).

% sums_iff_shift'
tff(fact_3314_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A,Nb: nat] :
          ( sums(A,F2,S)
         => sums(A,aa(nat,fun(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_split_initial_segment
tff(fact_3315_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S3: A,A4: set(nat),S5: A,F2: fun(nat,A)] :
          ( sums(A,G,S3)
         => ( finite_finite(nat,A4)
           => ( ( S5 = 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)),G),F2)),A4)) )
             => 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))),G),A4),F2),S5) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_3316_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A3: A] : powr(A,Xb,aa(A,A,uminus_uminus(A),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),powr(A,Xb,A3)) ) ).

% powr_minus_divide
tff(fact_3317_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xb) ) ) ).

% powr_neg_one
tff(fact_3318_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_3319_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_3320_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_3321_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_3322_ln__powr__bound,axiom,
    ! [Xb: real,A3: 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)),A3)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xb,A3)),A3)) ) ) ).

% ln_powr_bound
tff(fact_3323_ln__powr__bound2,axiom,
    ! [Xb: real,A3: 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)),A3)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),Xb),A3)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A3,A3)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_3324_tan__45,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = one_one(real) ).

% tan_45
tff(fact_3325_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_3326_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_3327_minus__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),aa(real,real,log(B2),Xb)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B2,Y)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_3328_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A,A3: A] :
          powr(A,Xb,A3) = $ite(Xb = zero_zero(A),zero_zero(A),exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,ln_ln(A),Xb)))) ) ).

% powr_def
tff(fact_3329_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => sums(A,aa(A,fun(nat,A),power_power(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_3330_power__half__series,axiom,
    sums(real,aTP_Lamp_ek(nat,real),one_one(real)) ).

% power_half_series
tff(fact_3331_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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,tan(real),X4)) ) ) ).

% lemma_tan_total
tff(fact_3332_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tan(real),Xb)) ) ) ).

% tan_gt_zero
tff(fact_3333_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),aa(real,real,aa(real,fun(real,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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( aa(real,real,tan(real),X4) = Y ) ) ).

% lemma_tan_total1
tff(fact_3334_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3335_tan__monotone_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3336_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xb)) ) ) ) ).

% tan_monotone
tff(fact_3337_tan__total,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X4)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( aa(real,real,tan(real),X4) = Y )
      & ! [Y5: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
            & ( aa(real,real,tan(real),Y5) = Y ) )
         => ( Y5 = X4 ) ) ) ).

% tan_total
tff(fact_3338_tan__minus__45,axiom,
    aa(real,real,tan(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% tan_minus_45
tff(fact_3339_tan__inverse,axiom,
    ! [Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,tan(real),Y)) = aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y)) ).

% tan_inverse
tff(fact_3340_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_3341_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3342_sums__if_H,axiom,
    ! [G: fun(nat,real),Xb: real] :
      ( sums(real,G,Xb)
     => sums(real,aTP_Lamp_el(fun(nat,real),fun(nat,real),G),Xb) ) ).

% sums_if'
tff(fact_3343_sums__if,axiom,
    ! [G: fun(nat,real),Xb: real,F2: fun(nat,real),Y: real] :
      ( sums(real,G,Xb)
     => ( sums(real,F2,Y)
       => sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_em(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% sums_if
tff(fact_3344_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),Nb))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),Nb))) ) ) ).

% powr_neg_numeral
tff(fact_3345_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_3346_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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
          & ( aa(real,real,tan(real),X4) = Y ) ) ) ).

% tan_total_pos
tff(fact_3347_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xb)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3348_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3349_tan__mono__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_3350_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),Xb))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3351_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( ( aa(real,real,tan(real),Xb) = Y )
         => ( aa(real,real,arctan,Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_3352_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_3353_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( aa(real,real,tan(real),aa(real,real,arctan,Y)) = Y ) ) ).

% arctan
tff(fact_3354_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3355_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3356_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3357_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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))),Z3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))
          & ( aa(real,real,tan(real),Z3) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_3358_cos__paired,axiom,
    ! [Xb: real] : sums(real,aTP_Lamp_en(real,fun(nat,real),Xb),cos(real,Xb)) ).

% cos_paired
tff(fact_3359_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tan(A),Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb))),one_one(A))) ) ).

% tan_half
tff(fact_3360_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_3361_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_3362_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_3363_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_3364_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M2: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),aa(nat,A,X6,M2)) )
         => topological_monoseq(A,X6) ) ) ).

% monoI2
tff(fact_3365_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_eo(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C2),Xb),suminf(A,aa(A,fun(nat,A),aTP_Lamp_eo(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))) ) ) ).

% diffs_equiv
tff(fact_3366_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,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))) ) ).

% arcosh_def
tff(fact_3367_sin__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,tan(real),Xb)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% sin_tan
tff(fact_3368_cos__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => ( cos(real,Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% cos_tan
tff(fact_3369_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))),comm_s3205402744901411588hammer(A,Z,Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Nb)) ) ).

% pochhammer_double
tff(fact_3370_real__sqrt__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,real,sqrt,Xb) = aa(real,real,sqrt,Y) )
    <=> ( Xb = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_3371_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_3372_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,sqrt,Xb) = zero_zero(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_3373_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% real_sqrt_less_iff
tff(fact_3374_real__sqrt__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% real_sqrt_le_iff
tff(fact_3375_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_3376_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_3377_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,one_one(nat)) = A3 ) ).

% pochhammer_1
tff(fact_3378_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_3379_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_3380_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_3381_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_3382_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_3383_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_3384_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: real,Y: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ).

% of_real_divide
tff(fact_3385_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_3386_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_3387_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_3388_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_3389_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_3390_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_3391_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_3392_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_3393_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_3394_real__sqrt__mult__self,axiom,
    ! [A3: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,A3)),aa(real,real,sqrt,A3)) = aa(real,real,abs_abs(real),A3) ).

% real_sqrt_mult_self
tff(fact_3395_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_3396_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_3397_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_3398_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_3399_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_3400_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_3401_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_3402_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_3403_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_3404_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_3405_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),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_3406_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),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_3407_real__sqrt__minus,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,Xb)) ).

% real_sqrt_minus
tff(fact_3408_real__sqrt__power,axiom,
    ! [Xb: real,K: nat] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xb)),K) ).

% real_sqrt_power
tff(fact_3409_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_3410_real__sqrt__divide,axiom,
    ! [Xb: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ).

% real_sqrt_divide
tff(fact_3411_real__sqrt__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_3412_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_3413_real__sqrt__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_3414_real__sqrt__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_3415_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( aa(real,real,sqrt,Xb) = zero_zero(real) )
       => ( Xb = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_3416_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_3417_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_3418_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Nb: nat,Mb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,Nb) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => ( comm_s3205402744901411588hammer(A,A3,Mb) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_3419_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Mb: nat,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,Mb) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => ( comm_s3205402744901411588hammer(A,A3,Nb) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_3420_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_3421_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,Xb: real] :
          ( ( Y != zero_zero(real) )
         => ( real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_3422_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,sqrt,Xb)) = aa(real,real,sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_3423_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_3424_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_3425_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_3426_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_3427_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_3428_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_3429_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Nb: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),Nb)) ) ).

% pochhammer_rec
tff(fact_3430_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X5: nat] : aa(nat,A,diffs(A,C2),X5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X5))),aa(nat,A,C2,aa(nat,nat,suc,X5))) ) ).

% diffs_def
tff(fact_3431_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Nb: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A3,Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ).

% pochhammer_Suc
tff(fact_3432_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_3433_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_3434_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Nb: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_3435_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,Nb) = zero_zero(A) )
        <=> ? [K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Nb)
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K2)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_3436_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_3437_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_3438_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_3439_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_3440_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_3441_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [X4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C2),X4))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)) ) ) ).

% termdiff_converges_all
tff(fact_3442_tan__60,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% tan_60
tff(fact_3443_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_3444_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_3445_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_3446_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),U)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_3447_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_3448_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_3449_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_3450_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_3451_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A3: real,C2: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),C2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),D2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_3452_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_3453_cos__45,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_45
tff(fact_3454_sin__45,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_45
tff(fact_3455_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_3456_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_3457_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_3458_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_3459_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_3460_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_3461_tan__30,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) ).

% tan_30
tff(fact_3462_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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_3463_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_3464_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_3465_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_3466_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xb)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% ln_sqrt
tff(fact_3467_cos__30,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_30
tff(fact_3468_sin__60,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_60
tff(fact_3469_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K)) ) ).

% pochhammer_minus'
tff(fact_3470_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_3471_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : cos(A,Xb) = sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Xb)) ) ).

% cos_sin_eq
tff(fact_3472_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : sin(A,Xb) = cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Xb)) ) ).

% sin_cos_eq
tff(fact_3473_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,K5: real,C2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),K5)
         => ( ! [X4: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X4)),K5)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C2),X4)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_es(A,fun(fun(nat,A),fun(nat,A)),Xb),C2)) ) ) ) ).

% termdiff_converges
tff(fact_3474_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_3475_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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_3476_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_3477_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_3478_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(real,real,sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_3479_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,uminus_uminus(A),sin(A,Xb)) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% minus_sin_cos_eq
tff(fact_3480_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),one_one(real)) ).

% cos_x_y_le_one
tff(fact_3481_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_3482_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_3483_cos__arctan,axiom,
    ! [Xb: real] : cos(real,aa(real,real,arctan,Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% cos_arctan
tff(fact_3484_sin__arctan,axiom,
    ! [Xb: real] : sin(real,aa(real,real,arctan,Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% sin_arctan
tff(fact_3485_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_3486_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_3487_arctan__half,axiom,
    ! [Xb: real] : aa(real,real,arctan,Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))) ).

% arctan_half
tff(fact_3488_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb))),semiring_char_0_fact(A,Nb)) ) ).

% fact_double
tff(fact_3489_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,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))) ) ).

% arsinh_def
tff(fact_3490_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_et(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_3491_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Nb: nat] :
          comm_s3205402744901411588hammer(A,A3,Nb) = $ite(Nb = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_eu(A,fun(nat,fun(A,A)),A3),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_3492_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_3493_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_3494_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ev(B,A)),A4) = one_one(A) ) ).

% prod.neutral_const
tff(fact_3495_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_3496_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ~ finite_finite(A,A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_3497_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A3: A,B2: fun(A,B)] :
          ( 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_ew(A,fun(fun(A,B),fun(A,B)),A3),B2)),S3) = $ite(member(A,A3,S3),aa(A,B,B2,A3),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_3498_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A3: A,B2: fun(A,B)] :
          ( 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_ex(A,fun(fun(A,B),fun(A,B)),A3),B2)),S3) = $ite(member(A,A3,S3),aa(A,B,B2,A3),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_3499_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( ~ member(A,Xb,A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert(A,Xb),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)) ) ) ) ) ).

% prod.insert
tff(fact_3500_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,Nb))),aa(nat,A,G,Nb)) ) ).

% prod.lessThan_Suc
tff(fact_3501_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Mb),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ).

% prod.cl_ivl_Suc
tff(fact_3502_arcsin__1,axiom,
    aa(real,real,arcsin,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% arcsin_1
tff(fact_3503_arcsin__minus__1,axiom,
    aa(real,real,arcsin,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% arcsin_minus_1
tff(fact_3504_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => ( aa(A,B,G,X4) = one_one(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_3505_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A4: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) != one_one(A) )
         => ~ ! [A5: B] :
                ( member(B,A5,A4)
               => ( aa(B,A,G,A5) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_3506_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A4: 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_ey(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),A4)) ) ).

% prod.distrib
tff(fact_3507_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),G: fun(B,A),A4: 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_ez(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A4) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4)) ) ).

% prod_dividef
tff(fact_3508_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A4: set(B),Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4)),Nb) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(nat,fun(B,A),aTP_Lamp_fa(fun(B,A),fun(nat,fun(B,A)),F2),Nb)),A4) ) ).

% prod_power_distrib
tff(fact_3509_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A3: A,A4: 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_bp(fun(B,A),fun(A,fun(B,A)),F2),A3)),A4),A3) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4),A3) ) ).

% mod_prod_eq
tff(fact_3510_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
         => aa(B,$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),F2),A4)) ) ) ).

% prod_nonneg
tff(fact_3511_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
         => aa(B,$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),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)) ) ) ).

% prod_mono
tff(fact_3512_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,X4)) )
         => aa(B,$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),F2),A4)) ) ) ).

% prod_pos
tff(fact_3513_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,X4)) )
         => aa(B,$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),F2),A4)) ) ) ).

% prod_ge_1
tff(fact_3514_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F2),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fb(fun(nat,A),fun(nat,fun(A,A)),F2),A3,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_3515_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_fc(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_fd(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A4) ) ) ) ).

% prod.inter_filter
tff(fact_3516_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F2: fun(B,nat),A4: set(B)] : aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F2),A4)) = 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_fe(A,fun(fun(B,nat),fun(B,A)),C2),F2)),A4) ) ).

% power_sum
tff(fact_3517_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ff(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_3518_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),one_one(B)) ) )
         => aa(B,$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),F2),A4)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_3519_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R2: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,one_one(A)),one_one(A))
         => ( ! [X15: A,Y15: A,X2: A,Y22: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R2,X15),X2)
                  & aa(A,$o,aa(A,fun(A,$o),R2,Y15),Y22) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(A,A,aa(A,fun(A,A),times_times(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y22)) )
           => ( finite_finite(B,S3)
             => ( ! [X4: B] :
                    ( member(B,X4,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R2,aa(B,A,H,X4)),aa(B,A,G,X4)) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3)) ) ) ) ) ) ).

% prod.related
tff(fact_3520_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert(A,Xb),A4)) = $ite(member(A,Xb,A4),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4))) ) ) ) ).

% prod.insert_if
tff(fact_3521_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T4: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T5: set(B),G: fun(A,C),H: fun(B,C)] :
          ( finite_finite(A,S5)
         => ( finite_finite(B,T4)
           => ( ! [A5: A] :
                  ( member(A,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S5))
                 => ( aa(B,A,I,aa(A,B,J,A5)) = A5 ) )
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S5))
                   => member(B,aa(A,B,J,A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),T4)) )
               => ( ! [B3: B] :
                      ( member(B,B3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),T4))
                     => ( aa(A,B,J,aa(B,A,I,B3)) = B3 ) )
                 => ( ! [B3: B] :
                        ( member(B,B3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),T4))
                       => member(A,aa(B,A,I,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S5)) )
                   => ( ! [A5: A] :
                          ( member(A,A5,S5)
                         => ( aa(A,C,G,A5) = one_one(C) ) )
                     => ( ! [B3: B] :
                            ( member(B,B3,T4)
                           => ( aa(B,C,H,B3) = one_one(C) ) )
                       => ( ! [A5: A] :
                              ( member(A,A5,S3)
                             => ( aa(B,C,H,aa(A,B,J,A5)) = aa(A,C,G,A5) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),T5) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_3522_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fg(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3523_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% prod.atLeastAtMost_rev
tff(fact_3524_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( finite_finite(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,I2)) )
               => aa(B,$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),F2),I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3525_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I2)) )
             => aa(B,$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),F2),I5)) ) ) ) ) ).

% less_1_prod
tff(fact_3526_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B4: set(A),A4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A4)
         => ( finite_finite(A,A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3527_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A4: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C5)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A4))
                   => ( aa(A,B,G,A5) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B4))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B4) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3528_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A4: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C5)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A4))
                   => ( aa(A,B,G,A5) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B4))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C5) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B4) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3529_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S3: set(A),G: fun(A,B)] :
          ( finite_finite(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T5)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S3))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T5) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3530_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S3: set(A),G: fun(A,B)] :
          ( finite_finite(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T5)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S3))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3531_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T5)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S3))
                 => ( aa(A,B,H,X4) = one_one(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S3)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T5) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3532_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T5)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S3))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S3)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T5) = 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_3533_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_3534_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3535_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3536_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fi(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3537_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fi(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3538_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_ce(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb))) ) ).

% fact_prod
tff(fact_3539_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( ! [I2: A] :
                ( member(A,I2,A4)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)) ) ) ) ) ).

% prod_mono_strict
tff(fact_3540_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( dvd_dvd(B,aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4))
          <=> ? [X: A] :
                ( member(A,X,A4)
                & dvd_dvd(B,aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)),aa(A,B,F2,X)) ) ) ) ) ).

% even_prod_iff
tff(fact_3541_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert(A,Xb),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_3542_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( member(A,Xb,A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_3543_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3544_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_3545_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_3546_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A3),Acc2,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A3),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_3547_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_3548_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A3: A,B2: fun(A,B),C2: fun(A,B)] :
          ( 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_fj(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B2),C2)),S3) = $ite(member(A,A3,S3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_3549_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_ce(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_3550_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B4: set(A),A4: set(A),F2: fun(A,B)] :
          ( finite_finite(A,B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
           => ( ! [B3: A] :
                  ( member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A4))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,B3)) )
             => ( ! [A5: A] :
                    ( member(A,A5,A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A5)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B4)) ) ) ) ) ) ).

% prod_mono2
tff(fact_3551_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A4: set(A),F2: fun(A,B),A3: A] :
          ( finite_finite(A,A4)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) = $ite(member(A,A3,A4),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(A,B,F2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)) ) ) ) ) ).

% prod_diff1
tff(fact_3552_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_3553_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Nb: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fk(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod
tff(fact_3554_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Nb: nat] : comm_s3205402744901411588hammer(A,A3,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fl(A,fun(nat,fun(nat,A)),A3),Nb)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)) ) ).

% pochhammer_prod_rev
tff(fact_3555_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Mb)),semiring_char_0_fact(nat,Nb)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ce(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_3556_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.in_pairs
tff(fact_3557_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fn(fun(nat,A),fun(nat,fun(A,A)),F2),A3,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3558_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Nb: nat] : comm_s3205402744901411588hammer(A,A3,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_fl(A,fun(nat,fun(nat,A)),A3),Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3559_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3560_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_3561_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ).

% arcsin_ubound
tff(fact_3562_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_3563_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,real,arcsin,sin(real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_3564_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_3565_le__arcsin__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,arcsin,Xb))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3566_arcsin__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xb)),Y)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3567_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),pi)
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3568_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_3569_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_3570_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_3571_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Nb) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_3572_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_fo(A,A),Nb,zero_zero(A)) ) ).

% of_nat_code
tff(fact_3573_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R: A,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fp(A,fun(nat,A),R)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Mb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Mb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,R),aa(nat,nat,suc,Mb))) ) ).

% gchoose_row_sum_weighted
tff(fact_3574_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_3575_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),one_one(nat)) = A3 ) ).

% gbinomial_1
tff(fact_3576_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_3577_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_3578_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_3579_prod__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( finite_finite(A,A4)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A4) = one_one(nat) )
      <=> ! [X: A] :
            ( member(A,X,A4)
           => ( aa(A,nat,F2,X) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_3580_prod__pos__nat__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( finite_finite(A,A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A4))
      <=> ! [X: A] :
            ( member(A,X,A4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3581_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_3582_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_3583_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_3584_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_3585_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_3586_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_3587_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_3588_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_3589_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_3590_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_3591_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_3592_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_3593_floor__divide__eq__div__numeral,axiom,
    ! [A3: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_divide_eq_div_numeral
tff(fact_3594_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_3595_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_3596_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_3597_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_3598_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_3599_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_3600_arccos__0,axiom,
    aa(real,real,arccos,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% arccos_0
tff(fact_3601_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_3602_floor__minus__divide__eq__div__numeral,axiom,
    ! [A3: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B2)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_3603_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_3604_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_3605_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),B2)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_3606_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_3607_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_3608_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_3609_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_3610_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_3611_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_3612_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_3613_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_3614_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K: int,L: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) ) ).

% floor_divide_of_int_eq
tff(fact_3615_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_3616_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_3617_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3618_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_3619_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_3620_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_3621_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Mb: nat,Nb: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)) ) ).

% floor_divide_of_nat_eq
tff(fact_3622_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_3623_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_3624_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_3625_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_3626_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A3),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3627_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),R),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_3628_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_3629_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_3630_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R))) ).

% real_of_int_floor_gt_diff_one
tff(fact_3631_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R))) ).

% real_of_int_floor_ge_diff_one
tff(fact_3632_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_bh(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_3633_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))) )
             => aa(int,$o,P,I4) ) ) ) ).

% floor_split
tff(fact_3634_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,Xb) = A3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),Xb)
            & aa(A,$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),A3)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_3635_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_3636_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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,A3)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% le_mult_floor
tff(fact_3637_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_3638_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_3639_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_3640_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_3641_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_3642_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_3643_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% Suc_times_gbinomial
tff(fact_3644_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_3645_floor__divide__real__eq__div,axiom,
    ! [B2: int,A3: real] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A3),aa(int,real,ring_1_of_int(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A3)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_3646_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Mb: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),Mb)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Mb)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3647_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( finite_finite(A,I5)
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,I2)) )
       => ( aa(real,real,ln_ln(real),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),F2),I5)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_fq(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_3648_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),Q2)),P2) ) ) ).

% floor_divide_lower
tff(fact_3649_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_3650_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_3651_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% gbinomial_factors
tff(fact_3652_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),K)),A3)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_3653_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),Nb)) ) ).

% gbinomial_index_swap
tff(fact_3654_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_3655_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),one_one(A))),Q2)) ) ) ).

% floor_divide_upper
tff(fact_3656_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_round(A,Xb) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% round_def
tff(fact_3657_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_3658_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_3659_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A3),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_3660_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_3661_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fr(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_3662_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fs(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_3663_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),Xb)) = K )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,aa(int,real,ring_1_of_int(real),K))),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),K),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_3664_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3665_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ).

% arccos_le_pi2
tff(fact_3666_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_3667_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_3668_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] :
          aa(nat,A,gbinomial(A,A3),K) = $ite(K = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_ft(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K))) ) ).

% gbinomial_code
tff(fact_3669_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] : aa(real,real,arccos,cos(real,Theta)) != aa(real,real,abs_abs(real),aa(real,real,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),K3)),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_3670_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fp(A,fun(nat,A),A3)),set_ord_atMost(nat,Mb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Mb)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_3671_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa(nat,nat,binomial(Nb),K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),one_one(nat)),Nb,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_3672_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_round(A,Xb) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,Xb)),archimedean_ceiling(A,Xb),archim6421214686448440834_floor(A,Xb)) ) ).

% round_altdef
tff(fact_3673_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_3674_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))),Nb)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb)))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_3675_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_atMost(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),K) ) ) ).

% atMost_iff
tff(fact_3676_binomial__n__n,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),Nb) = one_one(nat) ).

% binomial_n_n
tff(fact_3677_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_3678_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_3679_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_3680_binomial__n__0,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_3681_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_3682_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% sum.atMost_Suc
tff(fact_3683_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_3684_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% prod.atMost_Suc
tff(fact_3685_choose__one,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),one_one(nat)) = Nb ).

% choose_one
tff(fact_3686_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fu(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_3687_sum__choose__lower,axiom,
    ! [R: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fv(nat,fun(nat,nat),R)),set_ord_atMost(nat,Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R),Nb))),Nb) ).

% sum_choose_lower
tff(fact_3688_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fw(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_3689_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fw(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_3690_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
     => ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_3691_Suc__times__binomial,axiom,
    ! [K: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K)) ).

% Suc_times_binomial
tff(fact_3692_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).

% Suc_times_binomial_eq
tff(fact_3693_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_3694_choose__mult__lemma,axiom,
    ! [Mb: nat,R: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R)),Mb)) ).

% choose_mult_lemma
tff(fact_3695_binomial__le__pow,axiom,
    ! [R: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),R)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R)) ) ).

% binomial_le_pow
tff(fact_3696_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_fx(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_3697_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fy(nat,fun(nat,fun(nat,fun(nat,nat))),Mb),Nb),R)),set_ord_atMost(nat,R)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),R) ).

% vandermonde
tff(fact_3698_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_3699_binomial,axiom,
    ! [A3: 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),A3),B2)),Nb) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fz(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B2),Nb)),set_ord_atMost(nat,Nb)) ).

% binomial
tff(fact_3700_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_3701_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_3702_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_3703_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K)) ) ).

% zero_less_binomial
tff(fact_3704_Suc__times__binomial__add,axiom,
    ! [A3: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A3)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))),aa(nat,nat,suc,A3))) = 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),A3),B2))),A3)) ).

% Suc_times_binomial_add
tff(fact_3705_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K))),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_3706_choose__mult,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),Mb)),aa(nat,nat,binomial(Mb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K))) ) ) ) ).

% choose_mult
tff(fact_3707_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: 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),A3),B2)),Nb) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ga(A,fun(A,fun(nat,fun(nat,A))),A3),B2),Nb)),set_ord_atMost(nat,Nb)) ) ).

% binomial_ring
tff(fact_3708_binomial__absorb__comp,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_3709_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,B2: A,Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),Nb) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gb(A,fun(A,fun(nat,fun(nat,A))),A3),B2),Nb)),set_ord_atMost(nat,Nb)) ) ).

% pochhammer_binomial_sum
tff(fact_3710_choose__square__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_gc(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_3711_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,aa(num,nat,numeral_numeral(nat),K)),set_ord_atMost(nat,pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_3712_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,A3)),set_ord_lessThan(A,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3713_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_gd(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3714_binomial__absorption,axiom,
    ! [K: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_3715_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_3716_choose__linear__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ge(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_3717_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))),aa(nat,nat,binomial(Nb),K)) = semiring_char_0_fact(nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_3718_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_gf(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3719_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bz(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb))) ) ).

% sum.atMost_Suc_shift
tff(fact_3720_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,D2: fun(nat,A)] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(fun(nat,A),fun(A,fun(nat,A)),C2),X)),set_ord_atMost(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(fun(nat,A),fun(A,fun(nat,A)),D2),X)),set_ord_atMost(nat,Nb))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
             => ( aa(nat,A,C2,I4) = aa(nat,A,D2,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_3721_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: fun(nat,A),B4: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,A3,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A3),set_ord_atMost(nat,N))),B4)
           => summable(A,A3) ) ) ) ).

% bounded_imp_summable
tff(fact_3722_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fi(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb))) ) ).

% prod.atMost_Suc_shift
tff(fact_3723_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_3724_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_3725_binomial__mono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_mono
tff(fact_3726_binomial__antimono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),K)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_3727_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% binomial_maximum
tff(fact_3728_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% binomial_le_pow2
tff(fact_3729_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_3730_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_3731_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_3732_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_3733_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(fun(nat,A),fun(A,fun(nat,A)),C2),X)),set_ord_atMost(nat,Nb)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
             => ( aa(nat,A,C2,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_3734_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gh(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),set_ord_atMost(nat,Nb)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_3735_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_3736_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bz(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ).

% sum.atMost_shift
tff(fact_3737_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_atMost(nat,Mb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)))) ) ).

% sum_up_index_split
tff(fact_3738_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fi(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ).

% prod.atMost_shift
tff(fact_3739_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gi(A,fun(nat,A),A3)),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),A3),aa(nat,A,semiring_1_of_nat(A),Nb))),one_one(A))),Nb) ) ).

% gbinomial_parallel_sum
tff(fact_3740_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_gj(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gl(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ).

% sum.triangle_reindex_eq
tff(fact_3741_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_gj(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gn(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ).

% prod.triangle_reindex_eq
tff(fact_3742_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_3743_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_3744_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_strict_mono
tff(fact_3745_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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% central_binomial_odd
tff(fact_3746_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_3747_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_go(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_3748_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_gp(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_3749_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))) ) ) ) ).

% fact_binomial
tff(fact_3750_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))) ) ) ) ).

% binomial_fact
tff(fact_3751_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_3752_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_gq(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
        <=> ? [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
              & ( aa(nat,A,C2,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_3753_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_gq(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb))) ) ) ) ).

% polyfun_roots_finite
tff(fact_3754_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A3: A,Nb: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),A3)),set_ord_atMost(nat,Nb)) = zero_zero(A) )
         => ~ ! [B3: fun(nat,A)] :
                ~ ! [Z4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),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),A3)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),B3),Z4)),set_ord_lessThan(nat,Nb))) ) ) ).

% polyfun_linear_factor_root
tff(fact_3755_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Nb: nat,A3: A] :
        ? [B3: fun(nat,A)] :
        ! [Z4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),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),A3)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),B3),Z4)),set_ord_lessThan(nat,Nb)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),A3)),set_ord_atMost(nat,Nb))) ) ).

% polyfun_linear_factor
tff(fact_3756_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_3757_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_gs(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gl(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ).

% sum.triangle_reindex
tff(fact_3758_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_gs(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gn(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ).

% prod.triangle_reindex
tff(fact_3759_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_gt(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_gt(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gv(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_3760_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_gt(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_gt(fun(nat,A),fun(nat,real),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A3)),suminf(A,B2)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gv(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_3761_choose__two,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% choose_two
tff(fact_3762_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),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_cd(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ).

% sum.in_pairs_0
tff(fact_3763_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Mb: nat,A3: fun(nat,A),Nb: nat,B2: fun(nat,A),Xb: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),I2)
             => ( aa(nat,A,A3,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$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_gr(fun(nat,A),fun(A,fun(nat,A)),A3),Xb)),set_ord_atMost(nat,Mb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(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_gx(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A3),B2),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ) ) ) ).

% polynomial_product
tff(fact_3764_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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_fm(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ).

% prod.in_pairs_0
tff(fact_3765_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: A] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(fun(nat,A),fun(A,fun(nat,A)),C2),X)),set_ord_atMost(nat,Nb)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X: nat] :
                ( member(nat,X,set_or1337092689740270186AtMost(nat,one_one(nat),Nb))
               => ( aa(nat,A,C2,X) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_3766_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gy(A,fun(nat,A),A3)),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),A3),one_one(A))),Mb)) ) ).

% gbinomial_sum_lower_neg
tff(fact_3767_polynomial__product__nat,axiom,
    ! [Mb: nat,A3: 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,A3,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_gz(fun(nat,nat),fun(nat,fun(nat,nat)),A3),Xb)),set_ord_atMost(nat,Mb))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gz(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_hb(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A3),B2),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_3768_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_gt(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_gt(fun(nat,A),fun(nat,real),B2))
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gv(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A3)),suminf(A,B2))) ) ) ) ).

% Cauchy_product_sums
tff(fact_3769_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_3770_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P2)
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),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_3771_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P2)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_he(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),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_hf(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),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_3772_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat,A3: 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_hg(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A3),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),A3),Xb),Y)),set_ord_atMost(nat,Mb)) ) ).

% gbinomial_partial_sum_poly
tff(fact_3773_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,Z: A,A3: A] :
          ( aa(nat,$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) = A3 )
          <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_hi(nat,fun(A,fun(A,fun(nat,A))),Nb),Z),A3)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_3774_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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ).

% sum_gp0
tff(fact_3775_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hj(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_3776_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat,A3: 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_hg(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A3),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_hk(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A3),Xb),Y)),set_ord_atMost(nat,Mb)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_3777_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A3: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$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_gr(fun(nat,A),fun(A,fun(nat,A)),A3),Xb)),set_ord_atMost(nat,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),A3),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_hm(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A3),Xb),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff_alt
tff(fact_3778_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E: real,C2: fun(nat,A),Nb: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => ? [M7: real] :
            ! [Z4: A] :
              ( aa(real,$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_dj(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,Nb)))),aa(real,real,aa(real,fun(real,real),times_times(real),E),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z4)),aa(nat,nat,suc,Nb)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_3779_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A3: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$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_gr(fun(nat,A),fun(A,fun(nat,A)),A3),Xb)),set_ord_atMost(nat,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),A3),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_ho(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A3),Xb),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff
tff(fact_3780_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_hq(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_3781_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_hs(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_3782_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_hu(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_3783_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_dz(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_3784_inverse__eq__iff__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B2) )
        <=> ( A3 = B2 ) ) ) ).

% inverse_eq_iff_eq
tff(fact_3785_inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ).

% inverse_inverse_eq
tff(fact_3786_of__nat__id,axiom,
    ! [Nb: nat] : aa(nat,nat,semiring_1_of_nat(nat),Nb) = Nb ).

% of_nat_id
tff(fact_3787_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_3788_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_3789_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [Xb: A,A3: real,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,real_V8093663219630862766scaleR(A,A3),Y)) = aa(A,A,real_V8093663219630862766scaleR(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) ) ).

% mult_scaleR_right
tff(fact_3790_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [A3: real,Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),Y) = aa(A,A,real_V8093663219630862766scaleR(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) ) ).

% mult_scaleR_left
tff(fact_3791_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) ) ).

% inverse_mult_distrib
tff(fact_3792_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_3793_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_3794_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) ) ).

% inverse_divide
tff(fact_3795_inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ).

% inverse_minus_eq
tff(fact_3796_abs__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A3)) ) ).

% abs_inverse
tff(fact_3797_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_3798_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_3799_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% inverse_positive_iff_positive
tff(fact_3800_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_3801_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_3802_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_3803_scaleR__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,U: real,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,U),A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,real_V8093663219630862766scaleR(A,U),B2)) )
        <=> ( ( A3 = B2 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_3804_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: real,Y: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,real_V8093663219630862766scaleR(A,Xb),Y)),Nb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) ) ).

% scaleR_power
tff(fact_3805_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_3806_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_3807_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_3808_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),A3)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_3809_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),W)) ) ).

% inverse_eq_divide_numeral
tff(fact_3810_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),U)),A3)),aa(A,A,real_V8093663219630862766scaleR(A,U),A3)) = A3 ) ).

% scaleR_collapse
tff(fact_3811_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_3812_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),U)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),A3) ) ).

% scaleR_times
tff(fact_3813_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),W)),aa(num,real,numeral_numeral(real),V))),A3) ) ).

% inverse_scaleR_times
tff(fact_3814_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),aa(num,real,numeral_numeral(real),V))),A3) ) ).

% fraction_scaleR_times
tff(fact_3815_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)) = A3 ) ).

% scaleR_half_double
tff(fact_3816_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,Xb: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Y)) ) ).

% scaleR_right_distrib
tff(fact_3817_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_3818_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_3819_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B2) )
         => ( ( A3 != zero_zero(A) )
           => ( ( B2 != zero_zero(A) )
             => ( A3 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_3820_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
         => ( A3 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_3821_field__class_Ofield__inverse__zero,axiom,
    ! [A: $tType] :
      ( field(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% field_class.field_inverse_zero
tff(fact_3822_inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B2) )
         => ( A3 = B2 ) ) ) ).

% inverse_eq_imp_eq
tff(fact_3823_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),A3)),Nb) = aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) ) ).

% power_inverse
tff(fact_3824_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_3825_real__sqrt__inverse,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)) ).

% real_sqrt_inverse
tff(fact_3826_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Xb: A,C2: A,Y: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Mb),Xb)),C2) = Y )
          <=> ( Xb = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_3827_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Y: A,Xb: A,C2: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( Y = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Mb),Xb)),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),C2)) = Xb ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_3828_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_3829_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_3830_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_3831_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_3832_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_3833_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_3834_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_3835_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_3836_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_hv(A,fun(nat,A),Xb)) ) ).

% summable_exp_generic
tff(fact_3837_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3838_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3839_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3840_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3841_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_3842_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_3843_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_3844_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_3845_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R),real_V7770717601297561774m_norm(A,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xb))),aa(real,real,inverse_inverse(real),R)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_3846_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).

% inverse_less_imp_less
tff(fact_3847_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% less_imp_inverse_less
tff(fact_3848_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$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),A3) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_3849_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$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),A3)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_3850_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_3851_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_3852_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_3853_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ).

% positive_imp_inverse_positive
tff(fact_3854_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),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),A3)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_3855_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_3856_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_3857_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = B2 ) ) ) ).

% inverse_unique
tff(fact_3858_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B2)) ) ).

% field_class.field_divide_inverse
tff(fact_3859_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B2)) ) ).

% divide_inverse
tff(fact_3860_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A3) ) ).

% divide_inverse_commute
tff(fact_3861_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ).

% inverse_eq_divide
tff(fact_3862_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_3863_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_3864_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_3865_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A3)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_3866_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_3867_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),Xaa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Xb),Xaa)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xaa)) ) ).

% scaleR_left.add
tff(fact_3868_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,B2: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),B2)),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ).

% scaleR_left_distrib
tff(fact_3869_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,R),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,R)),Xb) ) ).

% scaleR_conv_of_real
tff(fact_3870_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R: real] : real_Vector_of_real(A,R) = aa(A,A,real_V8093663219630862766scaleR(A,R),one_one(A)) ) ).

% of_real_def
tff(fact_3871_divide__real__def,axiom,
    ! [Xb: real,Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y) = aa(real,real,aa(real,fun(real,real),times_times(real),Xb),aa(real,real,inverse_inverse(real),Y)) ).

% divide_real_def
tff(fact_3872_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_hv(A,fun(nat,A),Xb),exp(A,Xb)) ) ).

% exp_converges
tff(fact_3873_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X5: A] : exp(A,X5) = suminf(A,aTP_Lamp_hv(A,fun(nat,A),X5)) ) ).

% exp_def
tff(fact_3874_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_hw(A,fun(nat,real),Xb)) ) ).

% summable_norm_exp
tff(fact_3875_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% inverse_le_imp_le
tff(fact_3876_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% le_imp_inverse_le
tff(fact_3877_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$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),A3) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_3878_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$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),A3)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_3879_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_3880_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_less_inverse
tff(fact_3881_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_3882_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_3883_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),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),A3),B2)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_3884_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),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),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_3885_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A3)),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),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3886_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3887_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A3: real,C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C2)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_3888_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ) ) ).

% scaleR_right_mono
tff(fact_3889_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_3890_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_3891_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_3892_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A3: A,C2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_3893_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,Y: A,A3: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_3894_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),B2)),E)),C2)),D2) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3895_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)),E)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3896_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hx(A,fun(A,fun(nat,fun(nat,A))),Xb),Y),Nb)),set_ord_atMost(nat,Nb)) ) ) ) ).

% exp_series_add_commuting
tff(fact_3897_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_hy(A,fun(nat,A),Xb))) ) ).

% exp_first_term
tff(fact_3898_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_3899_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_3900_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_le_inverse
tff(fact_3901_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_3902_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_3903_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A3)),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),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_3904_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_3905_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_3906_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),B2)),zero_zero(A))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_3907_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: real,Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3908_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: real,C2: A,D2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D2)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3909_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xb: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_3910_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B2)) ) ) ).

% split_scaleR_pos_le
tff(fact_3911_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_3912_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_3913_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_3914_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B2)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_3915_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,A3: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),one_one(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),Xb) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_3916_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb) ) ).

% scaleR_2
tff(fact_3917_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D3),E2)
         => ( aa(real,$o,P,D3)
           => aa(real,$o,P,E2) ) )
     => ( ! [N: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => aa(real,$o,P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_3918_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero(nat) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N4)))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N4))),E) ) ) ).

% real_arch_inverse
tff(fact_3919_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D3),E2)
         => ( aa(real,$o,P,D3)
           => aa(real,$o,P,E2) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero(nat) )
           => aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N))) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => aa(real,$o,P,E) ) ) ) ).

% forall_pos_mono
tff(fact_3920_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,Xb)),Xb) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_3921_ln__inverse,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_inverse
tff(fact_3922_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,K: 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_hv(A,fun(nat,A),Xb)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_hz(A,fun(nat,fun(nat,A)),Xb),K))) ) ).

% exp_first_terms
tff(fact_3923_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_ia(A,fun(nat,A),Xb)) ) ).

% summable_exp
tff(fact_3924_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_3925_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_3926_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ib(A,fun(nat,A),Xb),sin(A,Xb)) ) ).

% sin_converges
tff(fact_3927_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X5: A] : sin(A,X5) = suminf(A,aTP_Lamp_ib(A,fun(nat,A),X5)) ) ).

% sin_def
tff(fact_3928_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ic(A,fun(nat,A),Xb),cos(A,Xb)) ) ).

% cos_converges
tff(fact_3929_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X5: A] : cos(A,X5) = suminf(A,aTP_Lamp_ic(A,fun(nat,A),X5)) ) ).

% cos_def
tff(fact_3930_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_id(A,fun(nat,real),Xb)) ) ).

% summable_norm_sin
tff(fact_3931_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_ie(A,fun(nat,real),Xb)) ) ).

% summable_norm_cos
tff(fact_3932_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_if(A,fun(nat,A),Xb))) ) ).

% exp_first_two_terms
tff(fact_3933_log__inverse,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,log(A3),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A3),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_3934_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ig(A,fun(nat,A),Xb),sin(A,Xb)) ) ).

% sin_minus_converges
tff(fact_3935_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ih(A,fun(nat,A),Xb),cos(A,Xb)) ) ).

% cos_minus_converges
tff(fact_3936_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_3937_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_3938_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_3939_tan__cot,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),Xb)) ).

% tan_cot
tff(fact_3940_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),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_3941_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),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_3942_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_3943_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ii(A,fun(nat,A),Xb),sinh(A,Xb)) ) ).

% sinh_converges
tff(fact_3944_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ij(A,fun(nat,A),Xb),cosh(A,Xb)) ) ).

% cosh_converges
tff(fact_3945_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),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,T3),sin(real,T3)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3946_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cot(real),Xb)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_3947_sinh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% sinh_real_less_iff
tff(fact_3948_sinh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sinh(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% sinh_real_pos_iff
tff(fact_3949_sinh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_3950_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_3951_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_3952_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_3953_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_3954_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_3955_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_3956_sinh__less__cosh__real,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),cosh(real,Xb)) ).

% sinh_less_cosh_real
tff(fact_3957_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_3958_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_3959_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : tanh(A,Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sinh(A,Xb)),cosh(A,Xb)) ) ).

% tanh_def
tff(fact_3960_cosh__real__pos,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cosh(real,Xb)) ).

% cosh_real_pos
tff(fact_3961_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_3962_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_3963_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_3964_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_3965_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_3966_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_3967_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_3968_complex__inverse,axiom,
    ! [A3: real,B2: real] : aa(complex,complex,inverse_inverse(complex),complex2(A3,B2)) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),A3),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),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_3969_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_3970_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X5: A] : aa(A,A,cot(A),X5) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,X5)),sin(A,X5)) ) ).

% cot_def
tff(fact_3971_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_3972_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) )
           => ( tanh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),tanh(A,Xb)),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),tanh(A,Xb)),tanh(A,Y)))) ) ) ) ) ).

% tanh_add
tff(fact_3973_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_3974_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : cosh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_3975_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : sinh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),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_3976_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_3977_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : cosh(A,Xb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% cosh_def
tff(fact_3978_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( cosh(real,aa(real,real,ln_ln(real),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,inverse_inverse(real),Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% cosh_ln_real
tff(fact_3979_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cot(real),Xb)) ) ) ).

% cot_gt_zero
tff(fact_3980_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sinh(A,Xb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% sinh_def
tff(fact_3981_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( sinh(real,aa(real,real,ln_ln(real),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),aa(real,real,inverse_inverse(real),Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% sinh_ln_real
tff(fact_3982_tan__cot_H,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Xb)) = aa(real,real,cot(real),Xb) ).

% tan_cot'
tff(fact_3983_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_ik(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_3984_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_il(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_3985_modulo__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),Mb)),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Mb,Nb))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa($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_3986_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)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_3987_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : sgn_sgn(A,sgn_sgn(A,A3)) = sgn_sgn(A,A3) ) ).

% sgn_sgn
tff(fact_3988_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_3989_sgn__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_one
tff(fact_3990_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_3991_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A3: A,B2: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sgn_sgn(A,A3)),sgn_sgn(A,B2)) ) ).

% sgn_divide
tff(fact_3992_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : sgn_sgn(A,aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),sgn_sgn(A,A3)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_3993_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Nb: nat] : sgn_sgn(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),sgn_sgn(A,A3)),Nb) ) ).

% power_sgn
tff(fact_3994_inverse__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),sgn_sgn(A,A3)) = sgn_sgn(A,A3) ) ).

% inverse_sgn
tff(fact_3995_sgn__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A3: A] : sgn_sgn(A,aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),sgn_sgn(A,A3)) ) ).

% sgn_inverse
tff(fact_3996_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),sgn_sgn(A,A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% sgn_greater
tff(fact_3997_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),sgn_sgn(A,A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_3998_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),sgn_sgn(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),sgn_sgn(A,B2)) ) ).

% divide_sgn
tff(fact_3999_nat__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_4000_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( sgn_sgn(A,A3) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_4001_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),sgn_sgn(A,A3)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_4002_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A3)),sgn_sgn(A,A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_4003_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_4004_nat__neg__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_4005_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),sgn_sgn(A,A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_4006_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : sgn_sgn(A,aa(A,A,abs_abs(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_4007_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_4008_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( sgn_sgn(A,A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_4009_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_4010_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : sgn_sgn(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = aa($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_4011_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_4012_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_4013_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_4014_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,Xb))),A3)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),A3)) ) ).

% nat_ceiling_le_eq
tff(fact_4015_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_4016_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_4017_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,Xb: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,A3)),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),A3),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_4018_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A3: 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,A3))
    <=> 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)),A3) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_4019_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A3: 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,A3))
    <=> 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)),A3) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_4020_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,Xb: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,A3)),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),A3),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_4021_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,Y: 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),sgn_sgn(A,Xb)),sgn_sgn(A,Y)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_4022_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A3) )
         => ( sgn_sgn(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = sgn_sgn(A,A3) ) ) ) ).

% same_sgn_sgn_add
tff(fact_4023_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B2: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A3)),sgn_sgn(A,B2)) ) ).

% sgn_mult
tff(fact_4024_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( sgn_sgn(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_4025_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( sgn_sgn(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_4026_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( sgn_sgn(A,B2) != sgn_sgn(A,A3) )
         => ( ( sgn_sgn(A,A3) != zero_zero(A) )
           => ( ( sgn_sgn(A,B2) != zero_zero(A) )
             => ( sgn_sgn(A,A3) = aa(A,A,uminus_uminus(A),sgn_sgn(A,B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4027_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_4028_nat__numeral__as__int,axiom,
    ! [X5: num] : aa(num,nat,numeral_numeral(nat),X5) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X5)) ).

% nat_numeral_as_int
tff(fact_4029_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),sgn_sgn(A,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_4030_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),sgn_sgn(A,A3)) = A3 ) ).

% abs_mult_sgn
tff(fact_4031_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A3)),aa(A,A,abs_abs(A),A3)) = A3 ) ).

% sgn_mult_abs
tff(fact_4032_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,Xb)),aa(A,A,abs_abs(A),Xb)) = Xb ) ).

% mult_sgn_abs
tff(fact_4033_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_4034_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A3) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_4035_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_4036_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_eq_sgn_abs
tff(fact_4037_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_4038_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_4039_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( sgn_sgn(A,A3) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% sgn_1_pos
tff(fact_4040_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          aa(A,A,abs_abs(A),sgn_sgn(A,A3)) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_4041_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_4042_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R)))) ) ).

% of_nat_ceiling
tff(fact_4043_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_4044_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_4045_nat__int__add,axiom,
    ! [A3: 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),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2) ).

% nat_int_add
tff(fact_4046_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ dvd_dvd(int,L,K)
       => ( sgn_sgn(int,modulo_modulo(int,K,L)) = sgn_sgn(int,L) ) ) ) ).

% sgn_mod
tff(fact_4047_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_4048_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_4049_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_4050_nat__plus__as__int,axiom,
    ! [X5: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X5),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_4051_nat__times__as__int,axiom,
    ! [X5: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X5),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_4052_nat__div__as__int,axiom,
    ! [X5: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X5),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_4053_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: 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_4054_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( sgn_sgn(A,A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_4055_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R)))),R) ) ) ).

% of_nat_floor
tff(fact_4056_zsgn__def,axiom,
    ! [I: 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_4057_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_4058_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_4059_split__nat,axiom,
    ! [P: fun(nat,$o),I: int] :
      ( aa(nat,$o,P,aa(int,nat,nat2,I))
    <=> ( ! [N4: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N4) )
           => aa(nat,$o,P,N4) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_4060_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] : aa(nat,$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,A3))),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),A3),B2)))) ) ).

% le_mult_nat_floor
tff(fact_4061_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K) ) ) ).

% le_nat_iff
tff(fact_4062_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_4063_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,V)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,V)),aa(int,int,abs_abs(int),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_4064_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_4065_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_4066_nat__div__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_4067_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_4068_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( dvd_dvd(int,L,K)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),sgn_sgn(int,L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_4069_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_4070_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_4071_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_4072_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_4073_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_4074_le__nat__floor,axiom,
    ! [Xb: nat,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Xb)),A3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(int,nat,nat2,archim6421214686448440834_floor(real,A3))) ) ).

% le_nat_floor
tff(fact_4075_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : modulo_modulo(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% mod_abs_eq_div_nat
tff(fact_4076_nat__take__bit__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_4077_take__bit__nat__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ) ).

% take_bit_nat_eq
tff(fact_4078_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(int,nat,nat2,K)),Nb)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ) ).

% bit_nat_iff
tff(fact_4079_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(int,L,K)))))) ) ).

% divide_int_def
tff(fact_4080_modulo__int__def,axiom,
    ! [K: int,L: int] :
      modulo_modulo(int,K,L) = $ite(
        L = zero_zero(int),
        K,
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($o,int,zero_neq_one_of_bool(int),~ dvd_dvd(int,L,K)))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))))) ) ).

% modulo_int_def
tff(fact_4081_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_4082_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_4083_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_4084_nat__abs__int__diff,axiom,
    ! [A3: 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),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) ).

% nat_abs_int_diff
tff(fact_4085_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_4086_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_4087_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K))) ) ).

% of_int_of_nat
tff(fact_4088_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( sgn_sgn(int,R) = sgn_sgn(int,L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R)),aa(int,int,abs_abs(int),L))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_4089_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3) )
               => ( ( sgn_sgn(int,R3) = sgn_sgn(int,A22) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),A22))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_4090_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K2: int] :
            ( ( A1 = K2 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K2) ) )
        | ? [L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K2 = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4) ) )
        | ? [R5: int,L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R5) )
            & ( sgn_sgn(int,R5) = sgn_sgn(int,L4) )
            & 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))
            & ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_4091_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa($o,int,zero_neq_one_of_bool(int),~ dvd_dvd(int,L,K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_4092_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(int,nat,nat2,K))
      <=> dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K) ) ) ).

% even_nat_iff
tff(fact_4093_pi__half,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) = the(real,aTP_Lamp_im(real,$o)) ).

% pi_half
tff(fact_4094_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_im(real,$o))) ).

% pi_def
tff(fact_4095_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,Nb)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Nb))))) ) ) ).

% powr_real_of_int
tff(fact_4096_divide__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),Mb))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,Nb,Mb)))))) ) ).

% divide_int_unfold
tff(fact_4097_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)
     => ( 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_4098_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = $ite(
        ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
        | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) ),
        aa(int,int,uminus_uminus(int),one_one(int)),
        $ite(
          K = zero_zero(int),
          L,
          $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),ord_max(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% or_int_unfold
tff(fact_4099_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero(real) )
     => ( aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xb)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),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_4100_setceilmax,axiom,
    ! [S: vEBT_VEBT,Mb: nat,Listy: list(vEBT_VEBT),Nb: nat] :
      ( vEBT_invar_vebt(S,Mb)
     => ( ! [X4: vEBT_VEBT] :
            ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
           => vEBT_invar_vebt(X4,Nb) )
       => ( ( Mb = aa(nat,nat,suc,Nb) )
         => ( ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
               => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X4)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) ) )
           => ( ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,S)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Mb))) )
             => ( aa(nat,int,semiring_1_of_nat(int),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,S),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))))) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Mb))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_4101_or_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: 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),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) ) ).

% or.right_idem
tff(fact_4102_or_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) ) ).

% or.left_idem
tff(fact_4103_or_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),A3) = A3 ) ).

% or.idem
tff(fact_4104_height__compose__list,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( member(vEBT_VEBT,Ta,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summarya),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))))) ) ).

% height_compose_list
tff(fact_4105_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A3) = A3 ) ).

% or.left_neutral
tff(fact_4106_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),zero_zero(A)) = A3 ) ).

% or.right_neutral
tff(fact_4107_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2)) ) ).

% take_bit_or
tff(fact_4108_max__ins__scaled,axiom,
    ! [Nb: nat,X14: vEBT_VEBT,Mb: nat,X13: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(nat),set(nat),insert(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))))) ).

% max_ins_scaled
tff(fact_4109_height__i__max,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Foo: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I))),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Foo),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))) ) ).

% height_i_max
tff(fact_4110_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_4111_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),K),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost
tff(fact_4112_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,A3: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),C2),set_ord_atMost(A,A3)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)) ) ).

% image_add_atMost
tff(fact_4113_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_4114_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_4115_max__idx__list,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Nb: nat,X14: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I)))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13)))))))) ) ).

% max_idx_list
tff(fact_4116_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_4117_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_4118_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : image(A,A,aTP_Lamp_in(A,fun(A,A),K),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_4119_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_4120_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_4121_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),Xb)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xb) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_4122_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),Xb)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xb) ) ) ) ) ) ).

% Max_less_iff
tff(fact_4123_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( image(A,A,aa(A,fun(A,A),times_times(A),D2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4124_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_4125_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_4126_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_4127_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_4128_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_4129_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( image(A,A,aTP_Lamp_io(A,fun(A,A),D2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4130_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_4131_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_4132_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_4133_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_4134_bit__or__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),Nb)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
        | aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_4135_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_4136_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_4137_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.left_commute
tff(fact_4138_or_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),A3) ) ).

% or.commute
tff(fact_4139_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.assoc
tff(fact_4140_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_or_eq
tff(fact_4141_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
            | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ).

% bit_or_iff
tff(fact_4142_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_4143_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_4144_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_4145_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_4146_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S3: set(A),F2: fun(A,B),K: B] :
          ( 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_ip(fun(A,B),fun(B,fun(A,B)),F2),K),S3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798349783984er_Max(B),image(A,B,F2,S3))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_4147_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_4148_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_4149_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) ) ) ) ).

% disjunctive_add
tff(fact_4150_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_4151_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( member(A,Xb,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) ) ) ) ).

% Max_ge
tff(fact_4152_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ! [Y4: A] :
                ( member(A,Y4,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
           => ( member(A,Xb,A4)
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = Xb ) ) ) ) ) ).

% Max_eqI
tff(fact_4153_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B4: set(A)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( ! [X4: A] :
                  ( member(A,X4,A4)
                 => ? [Xa: A] :
                      ( member(A,Xa,B4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,B4)
                   => ? [Xa: A] :
                        ( member(A,Xa,A4)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = aa(set(A),A,lattic643756798349783984er_Max(A),B4) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_4154_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite(A,A4)
         => ( member(A,A3,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) ) ) ) ).

% Max.coboundedI
tff(fact_4155_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Mb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = Mb )
            <=> ( member(A,Mb,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Mb) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4156_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A4))
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_4157_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Mb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( Mb = aa(set(A),A,lattic643756798349783984er_Max(A),A4) )
            <=> ( member(A,Mb,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Mb) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4158_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),Xb)
             => ! [A9: A] :
                  ( member(A,A9,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),Xb) ) ) ) ) ) ).

% Max.boundedE
tff(fact_4159_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),Xb) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),Xb) ) ) ) ) ).

% Max.boundedI
tff(fact_4160_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A4))
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_4161_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite(A,A4)
         => ( ! [B3: A] :
                ( member(A,B3,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,A3),A4)) = A3 ) ) ) ) ).

% Max_insert2
tff(fact_4162_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Uu,Dega,TreeLista,Summarya)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summarya),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))))) ).

% VEBT_internal.height.simps(2)
tff(fact_4163_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( ~ finite_finite(A,A4)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_4164_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: 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),A3),B2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
            & dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_or_iff
tff(fact_4165_sgn__real__def,axiom,
    ! [A3: real] :
      sgn_sgn(real,A3) = $ite(
        A3 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_4166_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Xb) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( Xb = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_4167_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( image(A,A,real_V8093663219630862766scaleR(A,C2),set_or1337092689740270186AtMost(A,Xb,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),Xb),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_4168_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_4169_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( finite_finite(A,B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),aa(set(A),A,lattic643756798349783984er_Max(A),B4)) ) ) ) ) ).

% Max.subset_imp
tff(fact_4170_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M6),N2)
         => ( ( M6 != bot_bot(set(A)) )
           => ( finite_finite(A,N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M6)),aa(set(A),A,lattic643756798349783984er_Max(A),N2)) ) ) ) ) ).

% Max_mono
tff(fact_4171_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xb) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xb = vEBT_Leaf((A5),(B3))
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Uu2,Deg,TreeList,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_4172_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = $ite(Nb = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_iq(nat,fun(nat,fun(nat,$o)),Mb),Nb)))) ).

% divide_nat_def
tff(fact_4173_sgn__power__injE,axiom,
    ! [A3: real,Nb: nat,Xb: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,A3)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A3)),Nb)) = Xb )
     => ( ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),Nb)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( A3 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_4174_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          image(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Xb)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4175_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_4176_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          image(A,A,aTP_Lamp_ir(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4177_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A3: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_is(A,fun(A,fun(A,A)),Mb),C2),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A3,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),A3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A3)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4178_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A3: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_it(A,fun(A,fun(A,A)),Mb),C2),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A3,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),A3)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A3)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4179_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A3: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_iu(A,fun(A,fun(A,A)),Mb),C2),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Mb)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Mb)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Mb)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Mb)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4180_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A3: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_iv(A,fun(A,fun(A,A)),Mb),C2),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Mb)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Mb)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Mb)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Mb)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4181_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).

% or_one_eq
tff(fact_4182_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).

% one_or_eq
tff(fact_4183_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_4184_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_4185_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)
            | ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% or_int_rec
tff(fact_4186_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_4187_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_4188_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,Ta: set(A)] : image(A,A,aa(A,fun(A,A),plus_plus(A),A3),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),A3),Ta)) ) ).

% translation_Compl
tff(fact_4189_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),Ta: set(A)] : image(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),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),A3),S)),image(A,A,aa(A,fun(A,A),plus_plus(A),A3),Ta)) ) ).

% translation_diff
tff(fact_4190_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A4: set(product_prod(A,B)),F2: 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),A3),B2),A4)
     => member(C,aa(B,C,aa(A,fun(B,C),F2,A3),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),F2),A4)) ) ).

% pair_imageI
tff(fact_4191_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_4192_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_4193_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_4194_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_4195_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_4196_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_4197_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_4198_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_4199_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_4200_None__notin__image__Some,axiom,
    ! [A: $tType,A4: set(A)] : ~ member(option(A),none(A),image(A,option(A),some(A),A4)) ).

% None_notin_image_Some
tff(fact_4201_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A4: set(A),F2: fun(nat,A),Nb: nat] :
      ( ( A4 = image(nat,A,F2,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),Nb))) )
     => finite_finite(A,A4) ) ).

% nat_seg_image_imp_finite
tff(fact_4202_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite(A,A4)
    <=> ? [N4: nat,F5: fun(nat,A)] : A4 = image(nat,A,F5,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),N4))) ) ).

% finite_conv_nat_seg_image
tff(fact_4203_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_4204_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_4205_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_4206_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_4207_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_4208_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_4209_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_4210_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_4211_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_4212_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_4213_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_4214_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 )
           => ! [M2: num] :
                ( ( Xaa = aa(num,num,bit0,M2) )
               => ( Y != aa(num,num,bit1,M2) ) ) )
         => ( ( ( Xb = one2 )
             => ! [M2: num] :
                  ( ( Xaa = aa(num,num,bit1,M2) )
                 => ( Y != aa(num,num,bit1,M2) ) ) )
           => ( ( ? [N: num] : Xb = aa(num,num,bit0,N)
               => ( ( Xaa = one2 )
                 => ( Y != aa(num,num,bit0,one2) ) ) )
             => ( ! [N: num] :
                    ( ( Xb = aa(num,num,bit0,N) )
                   => ! [M2: num] :
                        ( ( Xaa = aa(num,num,bit0,M2) )
                       => ( Y != bitM(bit_or_not_num_neg(N,M2)) ) ) )
               => ( ! [N: num] :
                      ( ( Xb = aa(num,num,bit0,N) )
                     => ! [M2: num] :
                          ( ( Xaa = aa(num,num,bit1,M2) )
                         => ( Y != aa(num,num,bit0,bit_or_not_num_neg(N,M2)) ) ) )
                 => ( ( ? [N: num] : Xb = aa(num,num,bit1,N)
                     => ( ( Xaa = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N: num] :
                          ( ( Xb = aa(num,num,bit1,N) )
                         => ! [M2: num] :
                              ( ( Xaa = aa(num,num,bit0,M2) )
                             => ( Y != bitM(bit_or_not_num_neg(N,M2)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xb = aa(num,num,bit1,N) )
                           => ! [M2: num] :
                                ( ( Xaa = aa(num,num,bit1,M2) )
                               => ( Y != bitM(bit_or_not_num_neg(N,M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_4215_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_4216_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_4217_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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% or_nat_rec
tff(fact_4218_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Mb),Nb) = $ite(
        Mb = zero_zero(nat),
        Nb,
        $ite(Nb = zero_zero(nat),Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% or_nat_unfold
tff(fact_4219_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_4220_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_4221_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_4222_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_4223_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_4224_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_4225_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),A3) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_4226_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A3) = A3 ) ).

% xor.left_neutral
tff(fact_4227_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),zero_zero(A)) = A3 ) ).

% xor.right_neutral
tff(fact_4228_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2)) ) ).

% take_bit_xor
tff(fact_4229_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_4230_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_4231_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_4232_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_4233_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_4234_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_4235_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_4236_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_4237_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_4238_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_4239_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_4240_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_4241_cis__pi__half,axiom,
    cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = imaginary_unit ).

% cis_pi_half
tff(fact_4242_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_4243_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_4244_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_4245_cis__minus__pi__half,axiom,
    cis(aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% cis_minus_pi_half
tff(fact_4246_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_4247_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_4248_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_4249_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B2)),Nb)
        <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
            <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ).

% bit_xor_iff
tff(fact_4250_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.left_commute
tff(fact_4251_xor_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),A3) ) ).

% xor.commute
tff(fact_4252_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.assoc
tff(fact_4253_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_xor_eq
tff(fact_4254_DeMoivre,axiom,
    ! [A3: real,Nb: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A3)),Nb) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A3)) ).

% DeMoivre
tff(fact_4255_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: 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),A3),B2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
          <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_xor_iff
tff(fact_4256_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Mb),Nb) = $ite(
        Mb = zero_zero(nat),
        Nb,
        $ite(Nb = zero_zero(nat),Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% xor_nat_unfold
tff(fact_4257_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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% xor_nat_rec
tff(fact_4258_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).

% xor_one_eq
tff(fact_4259_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).

% one_xor_eq
tff(fact_4260_Arg__minus__ii,axiom,
    arg(aa(complex,complex,uminus_uminus(complex),imaginary_unit)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% Arg_minus_ii
tff(fact_4261_Arg__ii,axiom,
    arg(imaginary_unit) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% Arg_ii
tff(fact_4262_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_iw(nat,fun(nat,complex),Nb),set_ord_lessThan(nat,Nb),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_by(nat,fun(complex,$o),Nb))) ) ).

% bij_betw_roots_unity
tff(fact_4263_csqrt__ii,axiom,
    csqrt(imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),imaginary_unit)),real_Vector_of_real(complex,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% csqrt_ii
tff(fact_4264_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A4: set(A),B4: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A3),A4,B4)
        <=> ( image(A,A,aa(A,fun(A,A),plus_plus(A),A3),A4) = B4 ) ) ) ).

% bij_betw_add
tff(fact_4265_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_4266_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_4267_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_4268_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),Nb)
    <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_4269_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_4270_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_4271_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T4: set(B),H: fun(A,B),S3: set(A),T5: set(B),G: fun(B,C)] :
          ( finite_finite(A,S5)
         => ( finite_finite(B,T4)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),T4))
             => ( ! [A5: A] :
                    ( member(A,A5,S5)
                   => ( aa(B,C,G,aa(A,B,H,A5)) = one_one(C) ) )
               => ( ! [B3: B] :
                      ( member(B,B3,T4)
                     => ( aa(B,C,G,B3) = one_one(C) ) )
                 => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ix(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T5) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_4272_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_4273_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_4274_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K) != ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% xor_int_rec
tff(fact_4275_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( sgn_sgn(complex,Z) = cis(Xb) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => ( arg(Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_4276_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( sgn_sgn(complex,Z) = cis(arg(Z)) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_4277_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
        K = aa(int,int,uminus_uminus(int),one_one(int)),
        aa(int,int,bit_ri4277139882892585799ns_not(int),L),
        $ite(
          L = aa(int,int,uminus_uminus(int),one_one(int)),
          aa(int,int,bit_ri4277139882892585799ns_not(int),K),
          $ite(
            K = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_4278_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_iy(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_4279_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_4280_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(list($o),int,aa(int,fun(list($o),int),aa(fun($o,int),fun(int,fun(list($o),int)),groups4207007520872428315er_sum($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_4281_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_4282_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_4283_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_4284_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_4285_bij__betw__Suc,axiom,
    ! [M6: set(nat),N2: set(nat)] :
      ( bij_betw(nat,nat,suc,M6,N2)
    <=> ( image(nat,nat,suc,M6) = N2 ) ) ).

% bij_betw_Suc
tff(fact_4286_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_4287_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_4288_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_4289_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_4290_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_4291_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_4292_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_4293_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_4294_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_4295_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_4296_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_4297_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_4298_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_4299_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_4300_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_4301_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_4302_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_4303_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_ri4277139882892585799ns_not(A),A3))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ).

% even_not_iff
tff(fact_4304_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_4305_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_4306_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_4307_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_not_numeral
tff(fact_4308_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_add
tff(fact_4309_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_4310_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_4311_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,Nb: nat] :
          ( member(A,A3,ring_1_Ints(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_4312_bit__not__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),Nb)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% bit_not_int_iff
tff(fact_4313_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_4314_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A,B2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2) ) ) ) ).

% take_bit_not_iff
tff(fact_4315_take__bit__not__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)) ) ).

% take_bit_not_take_bit
tff(fact_4316_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),K)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(int,A,ring_1_of_int(A),K)) ) ).

% of_int_not_eq
tff(fact_4317_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),B2) ) ).

% not_diff_distrib
tff(fact_4318_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),B2) ) ).

% not_add_distrib
tff(fact_4319_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_4320_and__eq__not__not__or,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),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),A3)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% and_eq_not_not_or
tff(fact_4321_or__eq__not__not__and,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),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),A3)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% or_eq_not_not_and
tff(fact_4322_or__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(int,int,bit_ri4277139882892585799ns_not(int),L))) ).

% or_int_def
tff(fact_4323_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] : finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_iz(A,fun(A,fun(A,$o)),A3),B2))) ) ).

% finite_int_segment
tff(fact_4324_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_4325_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),one_one(A)) ) ).

% not_eq_complement
tff(fact_4326_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_4327_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_4328_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_4329_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_4330_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A3: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N)
             => aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) ) ) ) ).

% disjunctive_diff
tff(fact_4331_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( member(A,A3,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)),A3)),A3) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_4332_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),bit_se2239418461657761734s_mask(A,Nb)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_4333_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_4334_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_4335_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_4336_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A3: int] :
          ( dvd_dvd(int,B2,A3)
         => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B2)),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_4337_xor__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),L))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),L)) ).

% xor_int_def
tff(fact_4338_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A] : finite_finite(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ja(A,fun(A,$o),A3))) ) ).

% finite_abs_int_segment
tff(fact_4339_not__int__div__2,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) ).

% not_int_div_2
tff(fact_4340_even__not__iff__int,axiom,
    ! [K: int] :
      ( dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K) ) ).

% even_not_iff_int
tff(fact_4341_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_4342_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_4343_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_4344_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_4345_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_4346_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K)),Nb)
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))),Nb) ) ).

% bit_minus_int_iff
tff(fact_4347_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_4348_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_4349_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_4350_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_4351_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_4352_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_4353_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_4354_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_4355_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_4356_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_4357_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_4358_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_4359_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_4360_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_4361_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_4362_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_4363_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_4364_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_4365_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_4366_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_4367_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),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,A3),Nb) ) ) ) ).

% bit_not_iff_eq
tff(fact_4368_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_4369_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A3: A] :
          ( ( archimedean_frac(A,Xb) = A3 )
        <=> ( member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A3),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_4370_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$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,A3)),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),A3),B2)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_4371_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$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),A3),B2)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_4372_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_4373_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_4374_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_4375_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)))) ) ).

% signed_take_bit_def
tff(fact_4376_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_4377_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_4378_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_4379_not__int__rec,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% not_int_rec
tff(fact_4380_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($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_4381_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,root(Nb),real_V7770717601297561774m_norm(complex,C2)))),cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),arg(C2)),aa(nat,real,semiring_1_of_nat(real),Nb))))),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_by(nat,fun(complex,$o),Nb)),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_jb(complex,fun(nat,fun(complex,$o)),C2),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_4382_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_4383_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_4384_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_4385_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_4386_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_4387_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A3: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_4388_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_4389_real__root__zero,axiom,
    ! [Nb: nat] : aa(real,real,root(Nb),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_4390_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3)) = aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),A3) ) ).

% push_bit_push_bit
tff(fact_4391_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ).

% push_bit_and
tff(fact_4392_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ).

% push_bit_or
tff(fact_4393_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ).

% push_bit_xor
tff(fact_4394_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_4395_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_4396_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_4397_root__0,axiom,
    ! [Xb: real] : aa(real,real,root(zero_zero(nat)),Xb) = zero_zero(real) ).

% root_0
tff(fact_4398_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) ) ).

% push_bit_Suc_numeral
tff(fact_4399_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_4400_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_4401_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_4402_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_4403_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_4404_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: 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),K))) = aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_4405_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_4406_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_4407_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_4408_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_4409_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_4410_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_4411_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_4412_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_4413_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,bit_se4730199178511100633sh_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) ) ).

% push_bit_numeral
tff(fact_4414_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_4415_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_4416_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),A3) = aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% push_bit_Suc
tff(fact_4417_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_4418_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3))
        <=> ( ( Nb != zero_zero(nat) )
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ) ).

% even_push_bit_iff
tff(fact_4419_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_4420_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [L: num,K: 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),K))) = 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,K)))) ) ).

% push_bit_minus_numeral
tff(fact_4421_real__root__minus,axiom,
    ! [Nb: nat,Xb: real] : aa(real,real,root(Nb),aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,root(Nb),Xb)) ).

% real_root_minus
tff(fact_4422_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3)) ) ).

% push_bit_minus
tff(fact_4423_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: int] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K)) ) ).

% push_bit_of_int
tff(fact_4424_real__root__commute,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa(real,real,root(Mb),aa(real,real,root(Nb),Xb)) = aa(real,real,root(Nb),aa(real,real,root(Mb),Xb)) ).

% real_root_commute
tff(fact_4425_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_4426_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_4427_push__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Nb),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K)) ).

% push_bit_nat_eq
tff(fact_4428_real__root__inverse,axiom,
    ! [Nb: nat,Xb: real] : aa(real,real,root(Nb),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,root(Nb),Xb)) ).

% real_root_inverse
tff(fact_4429_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ).

% push_bit_add
tff(fact_4430_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_4431_real__root__divide,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa(real,real,root(Nb),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ).

% real_root_divide
tff(fact_4432_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_4433_real__root__pos__pos__le,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_4434_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)) = 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),A3)) ) ).

% push_bit_take_bit
tff(fact_4435_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3)) = 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)),A3)) ) ).

% take_bit_push_bit
tff(fact_4436_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_4437_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_4438_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_4439_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_4440_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_4441_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_4442_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( sgn_sgn(real,aa(real,real,root(Nb),Xb)) = sgn_sgn(real,Xb) ) ) ).

% sgn_root
tff(fact_4443_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,Mb),K)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_4444_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Mb),Q2)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_4445_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_concat_bit(Nb,K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),L)) ).

% concat_bit_eq
tff(fact_4446_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Nb),A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_4447_concat__bit__def,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_concat_bit(Nb,K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),L)) ).

% concat_bit_def
tff(fact_4448_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : bit_se8732182000553998342ip_bit(A,Nb,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_4449_set__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),one_one(int))) ).

% set_bit_int_def
tff(fact_4450_flip__bit__int__def,axiom,
    ! [Nb: nat,K: int] : bit_se8732182000553998342ip_bit(int,Nb,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),one_one(int))) ).

% flip_bit_int_def
tff(fact_4451_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_4452_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(N2),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4453_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% sqrt_def
tff(fact_4454_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_4455_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,aa(A,fun(A,A),times_times(A),A3),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),A3)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% push_bit_double
tff(fact_4456_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_4457_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_4458_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Nb),A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),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_4459_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),one_one(int)))) ).

% unset_bit_int_def
tff(fact_4460_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_4461_real__root__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(N2),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4462_real__root__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N2),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_4463_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_4464_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_4465_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_4466_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_4467_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_4468_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_4469_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K) = aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% push_bit_int_def
tff(fact_4470_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_4471_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),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_4472_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: 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),A3)
         => ~ ! [B3: A] : A3 != aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B3) ) ) ).

% exp_dvdE
tff(fact_4473_real__root__increasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(N2),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_4474_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),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_4475_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),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_4476_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_4477_log__root,axiom,
    ! [Nb: nat,A3: 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)),A3)
       => ( aa(real,real,log(B2),aa(real,real,root(Nb),A3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(B2),A3)),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_root
tff(fact_4478_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_4479_ln__root,axiom,
    ! [Nb: nat,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Nb),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% ln_root
tff(fact_4480_split__root,axiom,
    ! [P: fun(real,$o),Nb: nat,Xb: real] :
      ( aa(real,$o,P,aa(real,real,root(Nb),Xb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ! [Y3: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y3)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y3)),Nb)) = Xb )
             => aa(real,$o,P,Y3) ) ) ) ) ).

% split_root
tff(fact_4481_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,real,root(Nb),Xb) = powr(real,Xb,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_4482_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A3: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A3) = $let(
            l: A,
            l:= aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A3),
            $ite(aa(nat,$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_4483_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_height_rel,Xb)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = zero_zero(nat) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Leaf((A5),(B3))) ) )
         => ~ ! [Uu2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,Deg,TreeList,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Node(Uu2,Deg,TreeList,Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_4484_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_4485_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ce(nat,nat)),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Ico_nat
tff(fact_4486_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_4487_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),A3) ) ).

% dual_order.refl
tff(fact_4488_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_4489_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_4490_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_4491_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_4492_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4493_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ finite_finite(A,set_or7035219750837199246ssThan(A,A3,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% infinite_Ico_iff
tff(fact_4494_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),K),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan
tff(fact_4495_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_4496_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : image(A,A,aTP_Lamp_in(A,fun(A,A),K),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_4497_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.op_ivl_Suc
tff(fact_4498_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.op_ivl_Suc
tff(fact_4499_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
            <=> ( ( A3 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4500_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
             => ( A3 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4501_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4502_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4503_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ finite_finite(A,set_or7035219750837199246ssThan(A,A3,B2)) ) ) ).

% infinite_Ico
tff(fact_4504_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Nb)
          & aa(nat,$o,P,M5) )
    <=> ? [X: nat] :
          ( member(nat,X,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less_eq
tff(fact_4505_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Nb)
         => aa(nat,$o,P,M5) )
    <=> ! [X: nat] :
          ( member(nat,X,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less_eq
tff(fact_4506_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_4507_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ff(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_4508_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A3: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A3 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),D2)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),set_or7035219750837199246ssThan(A,A3,B2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_4509_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A3: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A3 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),D2)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),set_or7035219750837199246ssThan(A,A3,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_4510_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P2)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Nb,P2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_4511_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,P2: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P2)
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,Mb,P2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,Mb,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,Nb,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_4512_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,P2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_4513_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_4514_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_4515_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X4: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_4516_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A3 = aa(B,A,F2,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X4: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_4517_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_4518_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_4519_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X4: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_subst2
tff(fact_4520_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X4: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_subst1
tff(fact_4521_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_4522_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
        <=> ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ) ).

% le_fun_def
tff(fact_4523_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G) ) ) ).

% le_funI
tff(fact_4524_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funE
tff(fact_4525_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funD
tff(fact_4526_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
           => ( A3 = B2 ) ) ) ) ).

% antisym
tff(fact_4527_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% dual_order.trans
tff(fact_4528_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => ( A3 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_4529_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% dual_order.eq_iff
tff(fact_4530_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B2: A] :
          ( ! [A5: A,B3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A5),B3) )
         => ( ! [A5: A,B3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B3),A5)
               => aa(A,$o,aa(A,fun(A,$o),P,A5),B3) )
           => aa(A,$o,aa(A,fun(A,$o),P,A3),B2) ) ) ) ).

% linorder_wlog
tff(fact_4531_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_4532_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% order.trans
tff(fact_4533_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_4534_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_4535_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_4536_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_4537_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_4538_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & ( B2 != A3 ) ) ) ) ).

% nle_le
tff(fact_4539_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xb: A] :
        ? [Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xb) ) ).

% lt_ex
tff(fact_4540_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xb: A] :
        ? [X_12: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X_12) ) ).

% gt_ex
tff(fact_4541_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_4542_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_4543_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).

% order.asym
tff(fact_4544_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_4545_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_4546_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A] :
          ( ! [X4: A] :
              ( ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X4)
                 => aa(A,$o,P,Y5) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A3) ) ) ).

% less_induct
tff(fact_4547_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_4548_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_4549_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% dual_order.asym
tff(fact_4550_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),A3) ) ).

% dual_order.irrefl
tff(fact_4551_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P,X_1)
        <=> ? [N4: A] :
              ( aa(A,$o,P,N4)
              & ! [M5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M5),N4)
                 => ~ aa(A,$o,P,M5) ) ) ) ) ).

% exists_least_iff
tff(fact_4552_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B2: A] :
          ( ! [A5: A,B3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A5),B3) )
         => ( ! [A5: A] : aa(A,$o,aa(A,fun(A,$o),P,A5),A5)
           => ( ! [A5: A,B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B3),A5)
                 => aa(A,$o,aa(A,fun(A,$o),P,A5),B3) )
             => aa(A,$o,aa(A,fun(A,$o),P,A3),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_4553_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% order.strict_trans
tff(fact_4554_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_4555_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% dual_order.strict_trans
tff(fact_4556_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( A3 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_4557_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( A3 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4558_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_4559_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_4560_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_4561_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).

% order_less_asym'
tff(fact_4562_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_4563_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A3 = aa(B,A,F2,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X4: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X4)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_4564_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X4: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_4565_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_4566_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X4: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X4)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_4567_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X4: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_4568_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_4569_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_4570_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_4571_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_4572_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_4573_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_4574_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4575_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4576_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_4577_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_4578_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_4579_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4580_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4581_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_4582_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Nb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% sum.last_plus
tff(fact_4583_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% prod.last_plus
tff(fact_4584_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ca(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Mb)) ) ) ) ).

% sum_Suc_diff'
tff(fact_4585_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_4586_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jc(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% sum.atLeastLessThan_rev
tff(fact_4587_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% prod.atLeastLessThan_rev
tff(fact_4588_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_je(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% sum.nat_group
tff(fact_4589_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% prod.nat_group
tff(fact_4590_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.head_if
tff(fact_4591_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.head_if
tff(fact_4592_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_bw(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4593_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4594_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Nb: nat] : comm_s3205402744901411588hammer(A,A3,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fk(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_prod
tff(fact_4595_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K: num] :
      set_or7035219750837199246ssThan(nat,Mb,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),pred_numeral(K)),aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_or7035219750837199246ssThan(nat,Mb,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_4596_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [N6: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M5)
                 => ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M5,N4)))),E4) ) ) ) ) ).

% summable_Cauchy
tff(fact_4597_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_4598_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M9: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M2)
                 => ! [N: nat] :
                      ( aa(nat,$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,M2)),aa(nat,A,X6,N)))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI
tff(fact_4599_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => ? [M7: nat] :
              ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M)
               => ! [N5: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N5)
                   => aa(real,$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,N5)))),E) ) ) ) ) ) ).

% CauchyD
tff(fact_4600_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A,K: nat] :
          ( sums(A,F2,S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_jg(fun(nat,A),fun(nat,fun(nat,A)),F2),K),S) ) ) ) ).

% sums_group
tff(fact_4601_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_jh(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% take_bit_sum
tff(fact_4602_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F2),G)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F2) ) ) ) ).

% less_fun_def
tff(fact_4603_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_4604_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_4605_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X4: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_4606_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X4: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_4607_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X4: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_4608_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X4: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X4)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_4609_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_4610_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_4611_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_4612_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( ( A3 != B2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_4613_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_4614_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_4615_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_4616_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_4617_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_4618_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% dual_order.strict_implies_order
tff(fact_4619_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% order.strict_implies_order
tff(fact_4620_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_4621_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% dual_order.strict_trans2
tff(fact_4622_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% dual_order.strict_trans1
tff(fact_4623_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4624_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
            | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4625_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_4626_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_4627_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% order.strict_iff_not
tff(fact_4628_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% order.strict_trans2
tff(fact_4629_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% order.strict_trans1
tff(fact_4630_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_4631_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
            | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_4632_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_4633_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_4634_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_4635_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_4636_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_4637_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_4638_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            | ( A3 = B2 ) ) ) ) ).

% nless_le
tff(fact_4639_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_4640_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_4641_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),bot_bot(A))
         => ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_4642_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),bot_bot(A))
        <=> ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_4643_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A3) ) ).

% bot.extremum
tff(fact_4644_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_4645_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( ( A3 != bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A3) ) ) ).

% bot.not_eq_extremum
tff(fact_4646_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xb: nat,Y: nat] :
      image(nat,nat,aTP_Lamp_ji(nat,fun(nat,nat),C2),set_or7035219750837199246ssThan(nat,Xb,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4647_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_4648_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_4649_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2),B2,A3) ) ).

% max_def
tff(fact_4650_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( semiring_char_0_fact(A,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K),Nb)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))) ) ) ) ).

% fact_split
tff(fact_4651_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jj(nat,fun(nat,fun(nat,A)),K),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_4652_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jk(A,fun(nat,fun(nat,A)),A3),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_4653_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),semiring_char_0_fact(A,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jl(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_4654_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A3),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jl(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_4655_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fs(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_4656_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_4657_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_mint_rel,Xb)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = $ite(
                      (A5),
                      aa(nat,option(nat),some(nat),zero_zero(nat)),
                      $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Leaf((A5),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_4658_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),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_jm(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_4659_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_maxt_rel,Xb)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      aa(nat,option(nat),some(nat),one_one(nat)),
                      $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Leaf((A5),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_4660_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A3: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$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,$o),ord_less_eq(A),aa(nat,A,A3,I2)),aa(nat,A,A3,J2)) ) )
         => ( ! [I2: nat,J2: nat] :
                ( aa(nat,$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_jn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),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,A3),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_4661_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A3: 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,A3,I2)),aa(nat,nat,A3,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_jo(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A3),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,A3),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_4662_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,Xb)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((A5),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Leaf((A5),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_4663_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,Xb)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((B3),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Leaf((A5),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_4664_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one(nat) )
             => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_4665_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_4666_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_4667_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ! [Uv2: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv2)) )
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) )
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_4668_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
     => vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq2
tff(fact_4669_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(Ta,D2)
     => vEBT_VEBT_valid(Ta,D2) ) ).

% valid_eq1
tff(fact_4670_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
    <=> vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq
tff(fact_4671_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D2: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv)),D2)
    <=> ( D2 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_4672_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_4673_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral(int) ).

% Code_Target_Int.positive_def
tff(fact_4674_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : re(csqrt(Z)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% csqrt.simps(1)
tff(fact_4675_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A3: real] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A3)) = re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A3)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_4676_csqrt_Ocode,axiom,
    ! [Z: complex] :
      csqrt(Z) = complex2(aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),
        aa(real,real,
          aa(real,fun(real,real),times_times(real),
            $ite(im(Z) = zero_zero(real),one_one(real),sgn_sgn(real,im(Z)))),
          aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% csqrt.code
tff(fact_4677_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),sgn_sgn(real,im(Z)))),
        aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% csqrt.simps(2)
tff(fact_4678_Complex__divide,axiom,
    ! [Xb: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),Y) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% Complex_divide
tff(fact_4679_Im__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( im(Xb) = zero_zero(real) )
     => ( im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xb),Nb)) = zero_zero(real) ) ) ).

% Im_power_real
tff(fact_4680_Re__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( im(Xb) = zero_zero(real) )
     => ( re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xb),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),Nb) ) ) ).

% Re_power_real
tff(fact_4681_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_4682_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(csqrt(Z)))
      | ( ( re(csqrt(Z)) = zero_zero(real) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(csqrt(Z))) ) ) ).

% csqrt_principal
tff(fact_4683_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A3: real] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A3)) = im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A3)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_4684_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_4685_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_4686_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_4687_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_4688_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_4689_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : re(aa(complex,complex,inverse_inverse(complex),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% inverse_complex.simps(1)
tff(fact_4690_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_4691_Re__divide,axiom,
    ! [Xb: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Re_divide
tff(fact_4692_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_4693_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_4694_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : im(aa(complex,complex,inverse_inverse(complex),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),im(Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% inverse_complex.simps(2)
tff(fact_4695_Im__divide,axiom,
    ! [Xb: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Im_divide
tff(fact_4696_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_4697_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) ) ) ).

% complex_unit_circle
tff(fact_4698_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : aa(complex,complex,inverse_inverse(complex),Xb) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),im(Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% inverse_complex.code
tff(fact_4699_Im__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( member(complex,R,real_Vector_Reals(complex))
     => ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R),Z)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),re(R))),im(Z))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% Im_Reals_divide
tff(fact_4700_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_4701_Re__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( member(complex,R,real_Vector_Reals(complex))
     => ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R),Z)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(R)),re(Z))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% Re_Reals_divide
tff(fact_4702_Reals__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,one_one(A),real_Vector_Reals(A)) ) ).

% Reals_1
tff(fact_4703_Reals__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A3: A,Nb: nat] :
          ( member(A,A3,real_Vector_Reals(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb),real_Vector_Reals(A)) ) ) ).

% Reals_power
tff(fact_4704_Reals__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_add
tff(fact_4705_Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_divide
tff(fact_4706_Reals__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_mult
tff(fact_4707_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_4708_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => ( ( B2 != zero_zero(A) )
             => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),real_Vector_Reals(A)) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_4709_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,complex),N2: nat,F2: fun(nat,A)] :
          ( summable(complex,G)
         => ( ! [N: nat] : member(complex,aa(nat,complex,G,N),real_Vector_Reals(complex))
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(nat,complex,G,N)))
             => ( ! [N: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),real_V7770717601297561774m_norm(complex,aa(nat,complex,G,N))) )
               => summable(A,F2) ) ) ) ) ) ).

% series_comparison_complex
tff(fact_4710_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),$o),aTP_Lamp_jp(nat,fun(list(A),fun(list(A),$o)),Nb),Xs)) ).

% set_n_lists
tff(fact_4711_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_4712_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_jq(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_4713_complex__cnj__power,axiom,
    ! [Xb: complex,Nb: nat] : cnj(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xb),Nb)) = aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cnj(Xb)),Nb) ).

% complex_cnj_power
tff(fact_4714_sgn__integer__code,axiom,
    ! [K: code_integer] :
      sgn_sgn(code_integer,K) = $ite(
        K = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),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_4715_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_4716_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_4717_Re__complex__div__gt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),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),A3),cnj(B2)))) ) ).

% Re_complex_div_gt_0
tff(fact_4718_Re__complex__div__lt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),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),A3),cnj(B2)))),zero_zero(real)) ) ).

% Re_complex_div_lt_0
tff(fact_4719_Im__complex__div__gt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),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),A3),cnj(B2)))) ) ).

% Im_complex_div_gt_0
tff(fact_4720_Im__complex__div__lt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),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),A3),cnj(B2)))),zero_zero(real)) ) ).

% Im_complex_div_lt_0
tff(fact_4721_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_4722_complex__div__gt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),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),A3),cnj(B2)))) )
      & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),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),A3),cnj(B2)))) ) ) ).

% complex_div_gt_0
tff(fact_4723_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_4724_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_4725_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_4726_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_4727_complex__div__cnj,axiom,
    ! [A3: complex,B2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),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_4728_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_4729_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),
            $ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_4730_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_4731_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( dvd_dvd(B,aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4))
          <=> dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_jr(set(A),fun(fun(A,B),fun(A,$o)),A4),F2)))) ) ) ) ).

% even_sum_iff
tff(fact_4732_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_ak(nat,fun(nat,$o)),Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_4733_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_aj(nat,fun(nat,$o)),Nb))) = aa(nat,nat,suc,Nb) ).

% card_Collect_le_nat
tff(fact_4734_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_js(A,fun(B,A),Y)),A4) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(set(B),nat,finite_card(B),A4)) ) ).

% prod_constant
tff(fact_4735_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A4: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_jt(A,fun(B,A),Y)),A4) = 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),A4))),Y) ) ).

% sum_constant
tff(fact_4736_card__Diff__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A),B4: set(A)] :
      ( member(A,A3,A4)
     => ( ~ member(A,A3,B4)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,A3),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_4737_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,abs_abs(code_integer),K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),K),K) ).

% abs_integer_code
tff(fact_4738_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_4739_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xaa),Xb) ) ).

% less_integer.abs_eq
tff(fact_4740_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),Xb)) ).

% divide_integer.abs_eq
tff(fact_4741_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B4: set(A),A4: set(B),R: fun(B,fun(A,$o))] :
      ( finite_finite(A,B4)
     => ( ! [A5: B] :
            ( member(B,A5,A4)
           => ? [B10: A] :
                ( member(A,B10,B4)
                & aa(A,$o,aa(B,fun(A,$o),R,A5),B10) ) )
       => ( ! [A12: B,A23: B,B3: A] :
              ( member(B,A12,A4)
             => ( member(B,A23,A4)
               => ( member(A,B3,B4)
                 => ( aa(A,$o,aa(B,fun(A,$o),R,A12),B3)
                   => ( aa(A,$o,aa(B,fun(A,$o),R,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_4742_card__insert__le,axiom,
    ! [A: $tType,A4: set(A),Xb: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),A4))) ).

% card_insert_le
tff(fact_4743_card__lists__length__eq,axiom,
    ! [A: $tType,A4: set(A),Nb: nat] :
      ( finite_finite(A,A4)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ah(set(A),fun(nat,fun(list(A),$o)),A4),Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A4)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_4744_card__eq__sum,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),nat,finite_card(A),A4) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_ju(A,nat)),A4) ).

% card_eq_sum
tff(fact_4745_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)) )
    <=> ? [X: A] :
          ( member(A,X,S3)
          & ? [Xa2: A] :
              ( member(A,Xa2,S3)
              & ( X != Xa2 )
              & ! [Xb4: A] :
                  ( member(A,Xb4,S3)
                 => ( ( Xb4 = X )
                    | ( Xb4 = Xa2 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_4746_card__ge__0__finite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4))
     => finite_finite(A,A4) ) ).

% card_ge_0_finite
tff(fact_4747_card__image__le,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B)] :
      ( finite_finite(A,A4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),image(A,B,F2,A4))),aa(set(A),nat,finite_card(A),A4)) ) ).

% card_image_le
tff(fact_4748_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Nb: nat,S3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),S3))
     => ~ ! [T2: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),S3)
           => ( ( aa(set(A),nat,finite_card(A),T2) = Nb )
             => ~ finite_finite(A,T2) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4749_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C5: nat] :
      ( ! [G4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),G4),F4)
         => ( finite_finite(A,G4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G4)),C5) ) )
     => ( finite_finite(A,F4)
        & aa(nat,$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_4750_card__seteq,axiom,
    ! [A: $tType,B4: set(A),A4: set(A)] :
      ( finite_finite(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B4)),aa(set(A),nat,finite_card(A),A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_4751_card__mono,axiom,
    ! [A: $tType,B4: set(A),A4: set(A)] :
      ( finite_finite(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4)) ) ) ).

% card_mono
tff(fact_4752_card__1__singletonE,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = one_one(nat) )
     => ~ ! [X4: A] : A4 != aa(set(A),set(A),insert(A,X4),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_4753_card__less__sym__Diff,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( finite_finite(A,A4)
     => ( finite_finite(A,B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4754_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_4755_card__le__sym__Diff,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( finite_finite(A,A4)
     => ( finite_finite(A,B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4756_psubset__card__mono,axiom,
    ! [A: $tType,B4: set(A),A4: set(A)] :
      ( finite_finite(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4)) ) ) ).

% psubset_card_mono
tff(fact_4757_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_jv(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,$o),aTP_Lamp_jw(set(nat),fun(nat,fun(nat,$o)),M6),I))) ) ) ).

% card_less_Suc2
tff(fact_4758_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_jv(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,$o),aTP_Lamp_jw(set(nat),fun(nat,fun(nat,$o)),M6),I))) ) ) ).

% card_less_Suc
tff(fact_4759_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_jw(set(nat),fun(nat,fun(nat,$o)),M6),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_4760_sum__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),A4: set(A)] : aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_jx(fun(A,nat),fun(A,nat),F2)),A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A4)),aa(set(A),nat,finite_card(A),A4)) ).

% sum_Suc
tff(fact_4761_subset__card__intvl__is__intvl,axiom,
    ! [A4: set(nat),K: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A4),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A4))))
     => ( A4 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_4762_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S3: set(A),T5: set(B),R2: fun(A,fun(B,$o)),K: nat] :
      ( finite_finite(A,S3)
     => ( finite_finite(B,T5)
       => ( ! [X4: B] :
              ( member(B,X4,T5)
             => ( 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_jy(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S3),R2),X4))) = K ) )
         => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_ka(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T5),R2)),S3) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T5)) ) ) ) ) ).

% sum_multicount
tff(fact_4763_real__of__card,axiom,
    ! [A: $tType,A4: set(A)] : aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),A4)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_kb(A,real)),A4) ).

% real_of_card
tff(fact_4764_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A4: set(A),K5: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K5),aa(A,B,F2,I2)) )
         => aa(B,$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),A4))),K5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)) ) ) ).

% sum_bounded_below
tff(fact_4765_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A4: set(A),F2: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),K5) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),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),A4))),K5)) ) ) ).

% sum_bounded_above
tff(fact_4766_card__gt__0__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4))
    <=> ( ( A4 != bot_bot(set(A)) )
        & finite_finite(A,A4) ) ) ).

% card_gt_0_iff
tff(fact_4767_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite(A,A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X: A] :
            ( member(A,X,A4)
           => ! [Xa2: A] :
                ( member(A,Xa2,A4)
               => ( X = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4768_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B4: set(B),F2: fun(A,B)] :
      ( finite_finite(A,A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),image(A,B,F2,A4))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B4)),aa(set(A),nat,finite_card(A),A4)) ) ) ).

% surj_card_le
tff(fact_4769_card__le__Suc__iff,axiom,
    ! [A: $tType,Nb: nat,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),A4))
    <=> ? [A6: A,B11: set(A)] :
          ( ( A4 = aa(set(A),set(A),insert(A,A6),B11) )
          & ~ member(A,A6,B11)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),B11))
          & finite_finite(A,B11) ) ) ).

% card_le_Suc_iff
tff(fact_4770_card__Diff1__le,axiom,
    ! [A: $tType,A4: set(A),Xb: A] : aa(nat,$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)),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4)) ).

% card_Diff1_le
tff(fact_4771_card__psubset,axiom,
    ! [A: $tType,B4: set(A),A4: set(A)] :
      ( finite_finite(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4) ) ) ) ).

% card_psubset
tff(fact_4772_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B4: set(A),A4: set(A)] :
      ( finite_finite(A,B4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))) ) ).

% diff_card_le_card_Diff
tff(fact_4773_card__lists__length__le,axiom,
    ! [A: $tType,A4: set(A),Nb: nat] :
      ( finite_finite(A,A4)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ai(set(A),fun(nat,fun(list(A),$o)),A4),Nb))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A4))),set_ord_atMost(nat,Nb)) ) ) ).

% card_lists_length_le
tff(fact_4774_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M6: set(A)] :
      ( 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_4775_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_4776_card__le__Suc__Max,axiom,
    ! [S3: set(nat)] :
      ( 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_4777_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N2)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_4778_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_ce(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_ce(nat,nat)),S3)) ).

% card_sum_le_nat_sum
tff(fact_4779_card__nth__roots,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_jb(complex,fun(nat,fun(complex,$o)),C2),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_4780_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,$o),set(complex),collect(complex),aTP_Lamp_by(nat,fun(complex,$o),Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_4781_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)) )
    <=> ? [X: A,Y3: A] :
          ( ( S3 = aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,Y3),bot_bot(set(A)))) )
          & ( X != Y3 ) ) ) ).

% card_2_iff
tff(fact_4782_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)) )
    <=> ? [X: A,Y3: A,Z2: A] :
          ( ( S3 = aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,Y3),aa(set(A),set(A),insert(A,Z2),bot_bot(set(A))))) )
          & ( X != Y3 )
          & ( Y3 != Z2 )
          & ( X != Z2 ) ) ) ).

% card_3_iff
tff(fact_4783_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(set(A),nat,finite_card(A),A4))
     => ( A4 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_4784_card__Diff1__less__iff,axiom,
    ! [A: $tType,A4: set(A),Xb: A] :
      ( aa(nat,$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)),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4))
    <=> ( finite_finite(A,A4)
        & member(A,Xb,A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4785_card__Diff2__less,axiom,
    ! [A: $tType,A4: set(A),Xb: A,Y: A] :
      ( finite_finite(A,A4)
     => ( member(A,Xb,A4)
       => ( member(A,Y,A4)
         => aa(nat,$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)),A4),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),A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4786_card__Diff1__less,axiom,
    ! [A: $tType,A4: set(A),Xb: A] :
      ( finite_finite(A,A4)
     => ( member(A,Xb,A4)
       => aa(nat,$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)),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4)) ) ) ).

% card_Diff1_less
tff(fact_4787_card__Diff__singleton__if,axiom,
    ! [A: $tType,A4: 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)),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))) = $ite(member(A,Xb,A4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A4)),one_one(nat)),aa(set(A),nat,finite_card(A),A4)) ).

% card_Diff_singleton_if
tff(fact_4788_card__Diff__singleton,axiom,
    ! [A: $tType,Xb: A,A4: set(A)] :
      ( member(A,Xb,A4)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),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),A4)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_4789_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_4790_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_4791_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B),Nb: B,K: nat] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),Nb) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),K)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),Nb)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Nb),K)) ) ) ) ) ).

% prod_le_power
tff(fact_4792_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A4: set(A),F2: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),K5) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),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),A4))),K5)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_4793_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A4: set(A),F2: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),K5),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A4)))) )
         => ( finite_finite(A,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),K5) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_4794_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_4795_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S3: set(A),R2: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( finite_finite(A,S3)
         => ( finite_finite(B,R2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,G,S3)),R2)
             => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_kc(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),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_ke(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S3),G),F2)),R2) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_4796_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A3: A,B2: fun(A,B),C2: B] :
          ( 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_kf(A,fun(fun(A,B),fun(B,fun(A,B))),A3),B2),C2)),S3) = $ite(member(A,A3,S3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A3)),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),S3)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(set(A),nat,finite_card(A),S3))) ) ) ) ).

% prod_gen_delta
tff(fact_4797_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_gq(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ).

% polyfun_roots_card
tff(fact_4798_sum__le__card__Max,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( finite_finite(A,A4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(A,nat,F2,A4)))) ) ).

% sum_le_card_Max
tff(fact_4799_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_4800_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_gq(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_gq(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ) ).

% polyfun_rootbound
tff(fact_4801_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( finite_finite(A,A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A4))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_kg(set(A),fun(nat,fun(list(A),$o)),A4),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ce(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),A4)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_4802_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(A),nat,finite_card(A),A4))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_kh(nat,fun(set(A),fun(list(A),$o)),K),A4))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ce(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),A4)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_4803_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_ki(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ) ).

% int_of_integer_code
tff(fact_4804_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,int,numeral_numeral(int),K) ).

% int_of_integer_numeral
tff(fact_4805_divide__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),divide_divide(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% divide_integer.rep_eq
tff(fact_4806_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_4807_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A4: set(A),Nb: nat] :
      ( finite_finite(A,A4)
     => finite_finite(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_kg(set(A),fun(nat,fun(list(A),$o)),A4),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4808_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_iff
tff(fact_4809_less__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Xb),Xaa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_integer.rep_eq
tff(fact_4810_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_4811_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( I4 != J3 )
               => ( aa(nat,A,nth(A,Xs),I4) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_4812_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_4813_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_4814_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,$o),ord_less(nat),X4),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X4) = Xb )
            & ! [Y5: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y5),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y5) = Xb ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4815_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A4: set(nat),B4: set(A)] :
      ( distinct(A,Xs)
     => ( ( A4 = set_ord_lessThan(nat,aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B4 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A4,B4) ) ) ) ).

% bij_betw_nth
tff(fact_4816_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A3: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ member(A,A3,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,A3)) ) ) ).

% distinct_list_update
tff(fact_4817_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_4818_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_kj(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% num_of_integer_code
tff(fact_4819_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_kk(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% nat_of_integer_code
tff(fact_4820_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),~ dvd_dvd(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)),K)) ).

% bit_cut_integer_def
tff(fact_4821_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_of_integer_code_post(3)
tff(fact_4822_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_4823_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,$o)),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o)),aTP_Lamp_kl(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% bit_cut_integer_code
tff(fact_4824_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,fun(A,A)),V: num,Nb: nat] :
      rec_nat(A,A3,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = $let(
        pv: nat,
        pv:= pred_numeral(V),
        aa(A,A,aa(nat,fun(A,A),F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb)),rec_nat(A,A3,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb))) ) ).

% rec_nat_add_eq_if
tff(fact_4825_case__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,A),V: num,Nb: nat] : case_nat(A,A3,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),Nb)) ).

% case_nat_add_eq_if
tff(fact_4826_old_Onat_Osimps_I7_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,fun(A,A)),Nat: nat] : rec_nat(A,F1,F22,aa(nat,nat,suc,Nat)) = aa(A,A,aa(nat,fun(A,A),F22,Nat),rec_nat(A,F1,F22,Nat)) ).

% old.nat.simps(7)
tff(fact_4827_old_Onat_Osimps_I6_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,fun(A,A))] : rec_nat(A,F1,F22,zero_zero(nat)) = F1 ).

% old.nat.simps(6)
tff(fact_4828_case__nat__numeral,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,A),V: num] : case_nat(A,A3,F2,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F2,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_4829_rec__nat__numeral,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,fun(A,A)),V: num] :
      rec_nat(A,A3,F2,aa(num,nat,numeral_numeral(nat),V)) = $let(
        pv: nat,
        pv:= pred_numeral(V),
        aa(A,A,aa(nat,fun(A,A),F2,pv),rec_nat(A,A3,F2,pv)) ) ).

% rec_nat_numeral
tff(fact_4830_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_km(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_4831_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X22: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X22)) = aa(nat,A,F22,X22) ).

% old.nat.simps(5)
tff(fact_4832_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_4833_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_kn(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_4834_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_ko(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_4835_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_4836_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_kp(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc1
tff(fact_4837_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_kq(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc2
tff(fact_4838_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_ce(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ).

% diff_Suc
tff(fact_4839_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_4840_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_4841_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xb: A,F2: fun(nat,A),Nb: nat] :
      case_nat(A,Xb,F2,Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_4842_old_Orec__nat__def,axiom,
    ! [A: $tType,X5: A,Xa: fun(nat,fun(A,A)),Xb2: nat] : rec_nat(A,X5,Xa,Xb2) = the(A,rec_set_nat(A,X5,Xa,Xb2)) ).

% old.rec_nat_def
tff(fact_4843_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_kr(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_zero(code_integer),
            aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
            product_apsnd(code_integer,code_integer,code_integer,uminus_uminus(code_integer),
              $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ks(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_4844_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_4845_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,B),Xb: A,Y: C] : product_apsnd(C,B,A,F2,aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Xb),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),aa(C,B,F2,Y)) ).

% apsnd_conv
tff(fact_4846_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,$o)),aTP_Lamp_kt(A,fun(B,fun(A,fun(B,$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_4847_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_ce(nat,nat),Nat) ).

% pred_def
tff(fact_4848_floor__real__def,axiom,
    ! [Xb: real] : archim6421214686448440834_floor(real,Xb) = the(int,aTP_Lamp_ku(real,fun(int,$o),Xb)) ).

% floor_real_def
tff(fact_4849_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_4850_floor__rat__def,axiom,
    ! [Xb: rat] : archim6421214686448440834_floor(rat,Xb) = the(int,aTP_Lamp_kv(rat,fun(int,$o),Xb)) ).

% floor_rat_def
tff(fact_4851_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_kw(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_4852_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: 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,K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_4853_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_4854_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Mb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3)) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),A3) ) ).

% drop_bit_drop_bit
tff(fact_4855_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),B2)) ) ).

% drop_bit_and
tff(fact_4856_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),B2)) ) ).

% drop_bit_or
tff(fact_4857_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A,B2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),B2)) ) ).

% drop_bit_xor
tff(fact_4858_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_4859_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_4860_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_4861_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_4862_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_4863_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_4864_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_4865_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: 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,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit0
tff(fact_4866_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: 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,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit1
tff(fact_4867_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: 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,K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_4868_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: 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,K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_4869_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: 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,K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_4870_sgn__rat__def,axiom,
    ! [A3: rat] :
      sgn_sgn(rat,A3) = $ite(
        A3 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A3),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_4871_less__eq__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),Xb),Y)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_rat_def
tff(fact_4872_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R)
     => ~ ! [S2: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),S2)
           => ! [T3: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T3)
               => ( R != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S2),T3) ) ) ) ) ).

% obtain_pos_sum
tff(fact_4873_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_4874_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_4875_abs__rat__def,axiom,
    ! [A3: rat] :
      aa(rat,rat,abs_abs(rat),A3) = $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),A3),zero_zero(rat)),aa(rat,rat,uminus_uminus(rat),A3),A3) ).

% abs_rat_def
tff(fact_4876_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) = A3 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_4877_drop__bit__push__bit__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] : aa(int,int,bit_se4197421643247451524op_bit(int,Mb),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K)) = 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)),K)) ).

% drop_bit_push_bit_int
tff(fact_4878_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3)) = 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)),A3)) ) ).

% take_bit_drop_bit
tff(fact_4879_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)) = 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),A3)) ) ).

% drop_bit_take_bit
tff(fact_4880_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_4881_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_4882_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: 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),A3))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)) = A3 ) ).

% bits_ident
tff(fact_4883_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A3 )
         => ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3) = A3 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_4884_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% drop_bit_half
tff(fact_4885_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% drop_bit_int_def
tff(fact_4886_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb)),A3) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% drop_bit_Suc
tff(fact_4887_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),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_4888_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3))
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_4889_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_4890_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,Mb: nat,A3: 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),A3))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),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_4891_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] :
          aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3) = $ite(Nb = zero_zero(nat),A3,aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% drop_bit_rec
tff(fact_4892_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q2),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_4893_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Mb: nat] :
      nat_prod_decode_aux(K,Mb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),K),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),Mb)),nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,K)))) ).

% prod_decode_aux.simps
tff(fact_4894_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_4895_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_4896_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,Nb),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K)) ).

% drop_bit_nat_eq
tff(fact_4897_normalize__denom__pos,axiom,
    ! [R: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% normalize_denom_pos
tff(fact_4898_drop__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,Nb),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% drop_bit_nat_def
tff(fact_4899_Suc__0__mod__numeral,axiom,
    ! [K: num] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_mod_numeral
tff(fact_4900_Suc__0__div__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_div_numeral
tff(fact_4901_finite__enumerate,axiom,
    ! [S3: set(nat)] :
      ( finite_finite(nat,S3)
     => ? [R3: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R3,set_ord_lessThan(nat,aa(set(nat),nat,finite_card(nat),S3)))
          & ! [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(set(nat),nat,finite_card(nat),S3))
             => member(nat,aa(nat,nat,R3,N5),S3) ) ) ) ).

% finite_enumerate
tff(fact_4902_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_4903_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_div_numeral
tff(fact_4904_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),K),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_mod_numeral
tff(fact_4905_fst__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(Mb,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) ).

% fst_divmod_nat
tff(fact_4906_snd__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(product_prod(nat,nat),nat,product_snd(nat,nat),divmod_nat(Mb,Nb)) = modulo_modulo(nat,Mb,Nb) ).

% snd_divmod_nat
tff(fact_4907_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,Nb)) ) ).

% one_div_numeral
tff(fact_4908_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_4909_snd__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: B,Y: A,A3: 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)) = A3 )
     => ( Y = A3 ) ) ).

% snd_eqD
tff(fact_4910_snd__conv,axiom,
    ! [B: $tType,A: $tType,X1: B,X22: 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),X22)) = X22 ).

% snd_conv
tff(fact_4911_fst__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,A3: 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)) = A3 )
     => ( Xb = A3 ) ) ).

% fst_eqD
tff(fact_4912_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X22: 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),X22)) = X1 ).

% fst_conv
tff(fact_4913_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_4914_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_4915_prod_Osplit__sel,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F2: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),Prod))
    <=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
       => aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F2,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel
tff(fact_4916_prod_Osplit__sel__asm,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F2: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),Prod))
    <=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
          & ~ aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F2,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_4917_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F2: fun(A,B),A4: set(A),Xb: A,Y: A] :
          ( strict_mono_on(A,B,F2,A4)
         => ( member(A,Xb,A4)
           => ( member(A,Y,A4)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_4918_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( strict_mono_on(A,B,F2,A4)
        <=> ! [R5: A,S6: A] :
              ( ( member(A,R5,A4)
                & member(A,S6,A4)
                & aa(A,$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,F2,R5)),aa(A,B,F2,S6)) ) ) ) ).

% strict_mono_on_def
tff(fact_4919_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [R3: A,S2: A] :
              ( member(A,R3,A4)
             => ( member(A,S2,A4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R3),S2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R3)),aa(A,B,F2,S2)) ) ) )
         => strict_mono_on(A,B,F2,A4) ) ) ).

% strict_mono_onI
tff(fact_4920_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A4: set(A),R: A,S: A] :
          ( strict_mono_on(A,B,F2,A4)
         => ( member(A,R,A4)
           => ( member(A,S,A4)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R)),aa(A,B,F2,S)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_4921_divides__aux__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Qr: product_prod(A,A)] :
          ( unique5940410009612947441es_aux(A,Qr)
        <=> ( aa(product_prod(A,A),A,product_snd(A,A),Qr) = zero_zero(A) ) ) ) ).

% divides_aux_def
tff(fact_4922_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% fst_divmod
tff(fact_4923_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_4924_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_4925_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_4926_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),P2: A,Q: fun(B,$o),Q2: B] :
      ( aa(A,$o,P,P2)
     => ( aa(B,$o,Q,Q2)
       => ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q2)))
          & aa(B,$o,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_4927_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_4928_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_4929_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_4930_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_4931_numeral__mod__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),Mb),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,Mb,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_4932_minus__numeral__mod__numeral,axiom,
    ! [Mb: num,Nb: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb)),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,Mb,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_4933_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R: int] :
      adjust_mod(L,R) = $ite(R = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R)) ).

% Divides.adjust_mod_def
tff(fact_4934_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Y)))))) ).

% bezw.simps
tff(fact_4935_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_4936_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Y)
     => ( bezw(Xb,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Y))))) ) ) ).

% bezw_non_0
tff(fact_4937_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Xaa)))))) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_4938_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Xb: A,Y: B,A3: product_prod(A,B)] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Xb),Y)
     => ( ( A3 = 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),A3)),aa(product_prod(A,B),B,product_snd(A,B),A3)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_4939_normalize__def,axiom,
    ! [P2: product_prod(int,int)] :
      normalize(P2) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
        $let(
          a2: int,
          a2:= aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a2)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(int,int),P2) = zero_zero(int),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),
          $let(
            a2: int,
            a2:= aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))),
            aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a2)) ) ) ) ).

% normalize_def
tff(fact_4940_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),one_one(A)) = one_one(A) ) ).

% gcd.bottom_right_bottom
tff(fact_4941_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),one_one(A)),A3) = one_one(A) ) ).

% gcd.bottom_left_bottom
tff(fact_4942_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_add2
tff(fact_4943_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Mb),Nb)),Nb) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_add1
tff(fact_4944_gcd__exp,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,Nb: nat,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),Nb) ) ).

% gcd_exp
tff(fact_4945_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: num,A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),A3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(num,A,numeral_numeral(A),Nb)),A3) ) ).

% gcd_neg_numeral_1
tff(fact_4946_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,Nb: num] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(num,A,numeral_numeral(A),Nb)) ) ).

% gcd_neg_numeral_2
tff(fact_4947_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2),one_one(A))
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_4948_gcd__pos__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Mb),Nb))
    <=> ( ( Mb != zero_zero(int) )
        | ( Nb != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_4949_gcd__neg__numeral__1__int,axiom,
    ! [Nb: num,Xb: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),Xb) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(num,int,numeral_numeral(int),Nb)),Xb) ).

% gcd_neg_numeral_1_int
tff(fact_4950_gcd__neg__numeral__2__int,axiom,
    ! [Xb: int,Nb: num] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),aa(num,int,numeral_numeral(int),Nb)) ).

% gcd_neg_numeral_2_int
tff(fact_4951_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,K: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),Mb)),Nb)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_add_mult
tff(fact_4952_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,K: A] : dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2)) ) ).

% gcd_dvd_prod
tff(fact_4953_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_4954_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_4955_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit2
tff(fact_4956_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit1
tff(fact_4957_gcd__le1__int,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),A3) ) ).

% gcd_le1_int
tff(fact_4958_gcd__le2__int,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),B2) ) ).

% gcd_le2_int
tff(fact_4959_gcd__non__0__int,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Y)
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),modulo_modulo(int,Xb,Y)) ) ) ).

% gcd_non_0_int
tff(fact_4960_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_4961_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,nat),P2: product_prod(A,B)] : basic_BNF_size_prod(A,B,F2,G,P2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,aa(product_prod(A,B),A,product_fst(A,B),P2))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P2)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_4962_nth__rotate1,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,rotate1(A,Xs)),Nb) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,Nb),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_4963_gcd__1__nat,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_4964_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb))
    <=> ( ( Mb != zero_zero(nat) )
        | ( Nb != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_4965_length__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),rotate1(A,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate1
tff(fact_4966_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4967_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_4968_gcd__le1__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),A3) ) ).

% gcd_le1_nat
tff(fact_4969_gcd__le2__nat,axiom,
    ! [B2: nat,A3: nat] :
      ( ( B2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),B2) ) ).

% gcd_le2_nat
tff(fact_4970_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_4971_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_4972_bezout__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [X4: nat,Y4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)) ) ).

% bezout_nat
tff(fact_4973_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A3: nat] :
    ? [X4: nat,Y4: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)),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),A3),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_4974_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_kx(nat,fun(nat,fun(nat,$o)),Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_4975_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun(nat,$o),Mb: nat] :
      ( ! [K3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K3)
         => aa(nat,$o,P,K3) )
     => ( ! [K3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),Nb)
           => ( ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),I3)
                 => aa(nat,$o,P,I3) )
             => aa(nat,$o,P,K3) ) )
       => aa(nat,$o,P,Mb) ) ) ).

% nat_descend_induct
tff(fact_4976_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q2: product_prod(A,B),F2: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P2: product_prod(A,B)] :
      ( ! [X4: A,Y4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) = Q2 )
         => ( aa(B,C,aa(A,fun(B,C),F2,X4),Y4) = aa(B,C,aa(A,fun(B,C),G,X4),Y4) ) )
     => ( ( P2 = Q2 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),P2) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q2) ) ) ) ).

% split_cong
tff(fact_4977_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Xaa) = Y )
     => ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),Xb,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),modulo_modulo(nat,Xb,Xaa))) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_4978_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_4979_xor__minus__numerals_I2_J,axiom,
    ! [K: int,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),neg_numeral_sub(int,Nb,one2))) ).

% xor_minus_numerals(2)
tff(fact_4980_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_4981_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_4982_sub__num__simps_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit0,K),aa(num,num,bit0,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(6)
tff(fact_4983_sub__num__simps_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K),aa(num,num,bit1,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(9)
tff(fact_4984_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_4985_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_4986_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_4987_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_4988_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_4989_sub__num__simps_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit0,K),aa(num,num,bit1,L)) = neg_numeral_dbl_dec(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(7)
tff(fact_4990_sub__num__simps_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K),aa(num,num,bit0,L)) = neg_numeral_dbl_inc(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(8)
tff(fact_4991_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_4992_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_4993_sub__num__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_sub(A,aa(num,num,bit1,K),one2) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,K)) ) ).

% sub_num_simps(5)
tff(fact_4994_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_4995_sub__num__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_sub(A,aa(num,num,bit0,K),one2) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% sub_num_simps(4)
tff(fact_4996_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_4997_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_4998_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_4999_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_5000_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_5001_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_5002_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_5003_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_5004_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_5005_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_5006_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),K) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),neg_numeral_sub(int,Nb,one2)),K)) ).

% xor_minus_numerals(1)
tff(fact_5007_neg__numeral__class_Osub__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,K,L) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) ) ).

% neg_numeral_class.sub_def
tff(fact_5008_quotient__of__denom__pos,axiom,
    ! [R: rat,P2: int,Q2: int] :
      ( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% quotient_of_denom_pos
tff(fact_5009_quotient__of__denom__pos_H,axiom,
    ! [R: rat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R))) ).

% quotient_of_denom_pos'
tff(fact_5010_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_5011_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_5012_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_5013_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_5014_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_5015_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_5016_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_5017_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P2),Q2)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_kz(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_5018_rat__floor__code,axiom,
    ! [P2: rat] : archim6421214686448440834_floor(rat,P2) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),divide_divide(int)),quotient_of(P2)) ).

% rat_floor_code
tff(fact_5019_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),one_one(rat)),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_5020_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).

% Frct_code_post(4)
tff(fact_5021_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xb: A,B2: B,A3: B] :
          ( nO_MATCH(A,B,Xb,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A3),B2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_5022_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xb: A,B2: B,A3: B] :
          ( nO_MATCH(A,B,Xb,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A3)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_5023_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,A3: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),A3)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_5024_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,A3: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),A3)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Y)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_5025_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,A3: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),A3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_5026_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,C2: B,A3: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A3),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_5027_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,A3: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),A3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A3),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_5028_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,C2: B,A3: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A3),B2)),C2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_5029_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_5030_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(num,rat,numeral_numeral(rat),K)),aa(num,rat,numeral_numeral(rat),L)) ).

% Frct_code_post(6)
tff(fact_5031_scale__left__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,C2: B,A3: real,B2: real] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),B2)),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_5032_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,C2: B,A3: real,B2: real] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),B2)),Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_5033_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),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_la(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_5034_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F2: fun(C,B),G: fun(D,A),Xb: product_prod(D,C)] : product_apsnd(C,B,A,F2,product_apfst(D,A,C,G,Xb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G,aa(product_prod(D,C),D,product_fst(D,C),Xb))),aa(C,B,F2,aa(product_prod(D,C),C,product_snd(D,C),Xb))) ).

% apsnd_apfst
tff(fact_5035_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F2: fun(C,A),G: fun(D,B),Xb: product_prod(C,D)] : product_apfst(C,A,B,F2,product_apsnd(D,B,C,G,Xb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,aa(product_prod(C,D),C,product_fst(C,D),Xb))),aa(D,B,G,aa(product_prod(C,D),D,product_snd(C,D),Xb))) ).

% apfst_apsnd
tff(fact_5036_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_5037_funpow__0,axiom,
    ! [A: $tType,F2: fun(A,A),Xb: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F2),Xb) = Xb ).

% funpow_0
tff(fact_5038_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F2: fun(C,A),Xb: C,Y: B] : product_apfst(C,A,B,F2,aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Xb),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,Xb)),Y) ).

% apfst_conv
tff(fact_5039_funpow__swap1,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat,Xb: A] : aa(A,A,F2,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),Xb)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),aa(A,A,F2,Xb)) ).

% funpow_swap1
tff(fact_5040_funpow__mod__eq,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A),Xb: A,Mb: nat] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),Xb) = Xb )
     => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),modulo_modulo(nat,Mb,Nb)),F2),Xb) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2),Xb) ) ) ).

% funpow_mod_eq
tff(fact_5041_funpow__mult,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),F2) ).

% funpow_mult
tff(fact_5042_bij__betw__funpow,axiom,
    ! [A: $tType,F2: fun(A,A),S3: set(A),Nb: nat] :
      ( bij_betw(A,A,F2,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),F2),S3,S3) ) ).

% bij_betw_funpow
tff(fact_5043_funpow__times__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [F2: fun(A,nat),Xb: A] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(A,nat,F2,Xb)),aa(A,fun(A,A),times_times(A),Xb)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(A,nat,F2,Xb))) ) ).

% funpow_times_power
tff(fact_5044_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A3) ) ).

% numeral_add_unfold_funpow
tff(fact_5045_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_5046_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),K) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% numeral_unfold_funpow
tff(fact_5047_relpowp__fun__conv,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Y)
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = Xb )
          & ( aa(nat,A,F5,Nb) = Y )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,F5,I4)),aa(nat,A,F5,aa(nat,nat,suc,I4))) ) ) ) ).

% relpowp_fun_conv
tff(fact_5048_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_5049_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_5050_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_5051_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q2: product_prod(A,B),F2: fun(C,A),P2: product_prod(C,B)] :
      ( ( Q2 = product_apfst(C,A,B,F2,P2) )
     => ~ ! [X4: C,Y4: B] :
            ( ( P2 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X4),Y4) )
           => ( Q2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_5052_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_aj(nat,fun(nat,$o)),aTP_Lamp_ak(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_5053_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_5054_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_5055_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S3: set(A),F2: fun(A,B)] :
          ( finite_finite(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ~ ? [X5: A] :
                  ( member(A,X5,S3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X5)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S3))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_5056_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S3: set(A),Y: A,F2: fun(A,B)] :
          ( finite_finite(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( member(A,Y,S3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S3))),aa(A,B,F2,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_5057_subrelI,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X4: A,Y4: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4),R)
         => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4),S) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),S) ) ).

% subrelI
tff(fact_5058_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$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_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(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))),R2),S3) ) ).

% pred_subset_eq2
tff(fact_5059_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( ! [X: 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),X),Xa2),R2)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa2),S3) )
    <=> ( R2 = S3 ) ) ).

% pred_equals_eq2
tff(fact_5060_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_5061_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_5062_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,P2: product_prod(A,B)] :
      ( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P2) )
    <=> ? [B5: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B5) ) ).

% eq_fst_iff
tff(fact_5063_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_5064_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_ld(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_5065_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_5066_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_5067_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_5068_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_5069_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_5070_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y4: nat] : Z != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X4),Y4)) ).

% eq_Abs_Integ
tff(fact_5071_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_5072_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_5073_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_5074_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_5075_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_5076_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_5077_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_5078_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_5079_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_5080_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_5081_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_5082_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_5083_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_5084_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_5085_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_5086_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_5087_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_le(nat,fun(nat,product_prod(nat,nat)))),Xb)) ).

% uminus_int.abs_eq
tff(fact_5088_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_5089_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),$o)),aTP_Lamp_lg(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_5090_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),$o)),aTP_Lamp_li(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_5091_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_5092_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_5093_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_lk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_5094_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_lm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_5095_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_5096_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P2: fun(A,B),I: A] :
          ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_aq(set(A),fun(fun(A,B),fun(A,$o)),I5),P2)))
         => ( groups1962203154675924110t_prod(A,B,P2,aa(set(A),set(A),insert(A,I),I5)) = $ite(member(A,I,I5),groups1962203154675924110t_prod(A,B,P2,I5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P2,I)),groups1962203154675924110t_prod(A,B,P2,I5))) ) ) ) ).

% prod.insert'
tff(fact_5097_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_5098_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_5099_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_5100_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1962203154675924110t_prod(B,A,G,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ln(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1962203154675924110t_prod(B,A,G,I5) ) ).

% prod.non_neutral'
tff(fact_5101_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_5102_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_5103_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_5104_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_5105_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite(A,I5)
         => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lo(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_5106_numeral__sqr,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),sqr(K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),K)) ) ).

% numeral_sqr
tff(fact_5107_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T5)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S3))
               => ( aa(A,B,G,X4) = one_one(B) ) )
           => ( ! [X4: A] :
                  ( member(A,X4,S3)
                 => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
             => ( groups1962203154675924110t_prod(A,B,G,T5) = groups1962203154675924110t_prod(A,B,H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_5108_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T5: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T5)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S3))
               => ( aa(A,B,H,I2) = one_one(B) ) )
           => ( ! [X4: A] :
                  ( member(A,X4,S3)
                 => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
             => ( groups1962203154675924110t_prod(A,B,G,S3) = groups1962203154675924110t_prod(A,B,H,T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_5109_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T5: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T5)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S3))
               => ( aa(A,B,G,X4) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,T5) = groups1962203154675924110t_prod(A,B,G,S3) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_5110_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T5: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T5)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S3))
               => ( aa(A,B,G,X4) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,S3) = groups1962203154675924110t_prod(A,B,G,T5) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_5111_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_5112_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_5113_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_aq(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_aq(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lo(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_5114_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: fun(B,A),I5: set(B)] :
          groups1962203154675924110t_prod(B,A,P2,I5) = $ite(finite_finite(B,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ln(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P2),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ln(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),one_one(A)) ) ).

% prod.G_def
tff(fact_5115_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_5116_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_5117_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_5118_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_5119_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_5120_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),$o)),aTP_Lamp_li(nat,fun(nat,fun(product_prod(nat,nat),$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_5121_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),$o)),aTP_Lamp_lg(nat,fun(nat,fun(product_prod(nat,nat),$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_5122_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_5123_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_le(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_5124_rat__floor__lemma,axiom,
    ! [A3: int,B2: int] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),fract(A3,B2))
      & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A3,B2)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),one_one(int)))) ) ).

% rat_floor_lemma
tff(fact_5125_Some__image__these__eq,axiom,
    ! [A: $tType,A4: set(option(A))] : image(A,option(A),some(A),these(A,A4)) = aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_lp(set(option(A)),fun(option(A),$o),A4)) ).

% Some_image_these_eq
tff(fact_5126_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_5127_these__insert__None,axiom,
    ! [A: $tType,A4: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),none(A)),A4)) = these(A,A4) ).

% these_insert_None
tff(fact_5128_these__insert__Some,axiom,
    ! [A: $tType,Xb: A,A4: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),aa(A,option(A),some(A),Xb)),A4)) = aa(set(A),set(A),insert(A,Xb),these(A,A4)) ).

% these_insert_Some
tff(fact_5129_these__image__Some__eq,axiom,
    ! [A: $tType,A4: set(A)] : these(A,image(A,option(A),some(A),A4)) = A4 ).

% these_image_Some_eq
tff(fact_5130_floor__Fract,axiom,
    ! [A3: int,B2: int] : archim6421214686448440834_floor(rat,fract(A3,B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) ).

% floor_Fract
tff(fact_5131_less__rat,axiom,
    ! [B2: int,D2: int,A3: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A3,B2)),fract(C2,D2))
        <=> 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),A3),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))) ) ) ) ).

% less_rat
tff(fact_5132_Rat__induct__pos,axiom,
    ! [P: fun(rat,$o),Q2: rat] :
      ( ! [A5: int,B3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
         => aa(rat,$o,P,fract(A5,B3)) )
     => aa(rat,$o,P,Q2) ) ).

% Rat_induct_pos
tff(fact_5133_in__these__eq,axiom,
    ! [A: $tType,Xb: A,A4: set(option(A))] :
      ( member(A,Xb,these(A,A4))
    <=> member(option(A),aa(A,option(A),some(A),Xb),A4) ) ).

% in_these_eq
tff(fact_5134_Fract__coprime,axiom,
    ! [A3: int,B2: int] : fract(aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2))) = fract(A3,B2) ).

% Fract_coprime
tff(fact_5135_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_5136_rat__number__expand_I3_J,axiom,
    ! [K: num] : aa(num,rat,numeral_numeral(rat),K) = fract(aa(num,int,numeral_numeral(int),K),one_one(int)) ).

% rat_number_expand(3)
tff(fact_5137_these__not__empty__eq,axiom,
    ! [A: $tType,B4: set(option(A))] :
      ( ( these(A,B4) != bot_bot(set(A)) )
    <=> ( ( B4 != bot_bot(set(option(A))) )
        & ( B4 != aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_5138_these__empty__eq,axiom,
    ! [A: $tType,B4: set(option(A))] :
      ( ( these(A,B4) = bot_bot(set(A)) )
    <=> ( ( B4 = bot_bot(set(option(A))) )
        | ( B4 = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_5139_zero__less__Fract__iff,axiom,
    ! [B2: int,A3: 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),zero_zero(rat)),fract(A3,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A3) ) ) ).

% zero_less_Fract_iff
tff(fact_5140_Fract__less__zero__iff,axiom,
    ! [B2: int,A3: 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(A3,B2)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),zero_zero(int)) ) ) ).

% Fract_less_zero_iff
tff(fact_5141_Fract__less__one__iff,axiom,
    ! [B2: int,A3: 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(A3,B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B2) ) ) ).

% Fract_less_one_iff
tff(fact_5142_one__less__Fract__iff,axiom,
    ! [B2: int,A3: 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(A3,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),A3) ) ) ).

% one_less_Fract_iff
tff(fact_5143_Option_Othese__def,axiom,
    ! [A: $tType,A4: set(option(A))] : these(A,A4) = image(option(A),A,the2(A),aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_lp(set(option(A)),fun(option(A),$o),A4))) ).

% Option.these_def
tff(fact_5144_Fract__le__zero__iff,axiom,
    ! [B2: int,A3: 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(A3,B2)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int)) ) ) ).

% Fract_le_zero_iff
tff(fact_5145_zero__le__Fract__iff,axiom,
    ! [B2: int,A3: 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),zero_zero(rat)),fract(A3,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3) ) ) ).

% zero_le_Fract_iff
tff(fact_5146_Fract__le__one__iff,axiom,
    ! [B2: int,A3: 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(A3,B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),B2) ) ) ).

% Fract_le_one_iff
tff(fact_5147_one__le__Fract__iff,axiom,
    ! [B2: int,A3: 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(A3,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A3) ) ) ).

% one_le_Fract_iff
tff(fact_5148_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_5149_rat__number__expand_I5_J,axiom,
    ! [K: num] : aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K)) = fract(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% rat_number_expand(5)
tff(fact_5150_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_ld(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_5151_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_lm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_5152_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_lk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_5153_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))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),aTP_Lamp_lq(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_5154_positive__rat,axiom,
    ! [A3: int,B2: int] :
      ( aa(rat,$o,positive,fract(A3,B2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B2)) ) ).

% positive_rat
tff(fact_5155_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,P,A3)
           => ( ~ aa(A,$o,P,B2)
             => ? [C3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
                  & ! [X5: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X5)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),C3) )
                     => aa(A,$o,P,X5) )
                  & ! [D6: A] :
                      ( ! [X4: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X4)
                            & aa(A,$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),C3) ) ) ) ) ) ) ).

% complete_interval
tff(fact_5156_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_5157_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_5158_take__bit__num__simps_I5_J,axiom,
    ! [R: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_5159_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)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),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_5160_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,aa(option(C),B,aa(fun(C,B),fun(option(C),B),aa(B,fun(fun(C,B),fun(option(C),B)),case_option(B,C),F1),F22),Option)) = aa(option(C),A,aa(fun(C,A),fun(option(C),A),aa(A,fun(fun(C,A),fun(option(C),A)),case_option(A,C),aa(B,A,H,F1)),aa(fun(C,B),fun(C,A),aTP_Lamp_lr(fun(B,A),fun(fun(C,B),fun(C,A)),H),F22)),Option) ).

% option.case_distrib
tff(fact_5161_option_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),X22: B] : aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),case_option(A,B),F1),F22),aa(B,option(B),some(B),X22)) = aa(B,A,F22,X22) ).

% option.simps(5)
tff(fact_5162_option_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,A)] : aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),case_option(A,B),F1),F22),none(B)) = F1 ).

% option.simps(4)
tff(fact_5163_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_ls(nat,option(num)),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_5164_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_5165_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: num,Q2: num] :
          ( ( bit_take_bit_num(Mb,Nb) = aa(num,option(num),some(num),Q2) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_5166_option_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),Option: option(B)] :
      aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),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_5167_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A3: A] :
        ? [B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% ex_gt_or_lt
tff(fact_5168_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_5169_option_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),case_option(A,B),F1),F22),Option))
    <=> ~ ( ( ( Option = none(B) )
            & ~ aa(A,$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_5170_option_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),case_option(A,B),F1),F22),Option))
    <=> ( ( ( Option = none(B) )
         => aa(A,$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_5171_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_5172_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)))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(Mb,bitM(Nb))) ).

% and_minus_numerals(3)
tff(fact_5173_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)) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(Mb,bitM(Nb))) ).

% and_minus_numerals(7)
tff(fact_5174_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)))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),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_5175_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),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_5176_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)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_lt(num,option(num))),bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_5177_take__bit__num__simps_I7_J,axiom,
    ! [R: num,Mb: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit1,Mb)) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(pred_numeral(R),Mb))) ).

% take_bit_num_simps(7)
tff(fact_5178_take__bit__num__simps_I6_J,axiom,
    ! [R: num,Mb: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit0,Mb)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_lt(num,option(num))),bit_take_bit_num(pred_numeral(R),Mb)) ).

% take_bit_num_simps(6)
tff(fact_5179_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)) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),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_5180_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)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_lu(num,option(num))),bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(8)
tff(fact_5181_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_5182_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),aa($o,fun(fun(A,$o),fun(option(A),$o)),case_option($o,A),$false),aTP_Lamp_lv(A,$o)),Option) ) ).

% option.disc_eq_case(2)
tff(fact_5183_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),aa($o,fun(fun(A,$o),fun(option(A),$o)),case_option($o,A),$true),aTP_Lamp_lw(A,$o)),Option) ) ).

% option.disc_eq_case(1)
tff(fact_5184_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_5185_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_5186_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_5187_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_lx(num,fun(nat,option(num)),Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_5188_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),Xb: option(A)] :
      ( aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),aa($o,fun(fun(A,$o),fun(option(A),$o)),case_option($o,A),(P)),Q),Xb)
     => ( ( ( Xb = none(A) )
         => ~ (P) )
       => ~ ! [Y4: A] :
              ( ( Xb = aa(A,option(A),some(A),Y4) )
             => ~ aa(A,$o,Q,Y4) ) ) ) ).

% case_optionE
tff(fact_5189_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_5190_and__not__num__eq__Some__iff,axiom,
    ! [Mb: num,Nb: num,Q2: num] :
      ( ( bit_and_not_num(Mb,Nb) = aa(num,option(num),some(num),Q2) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))) = aa(num,int,numeral_numeral(int),Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_5191_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_ly(num,fun(nat,option(num)),Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_5192_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_5193_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))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(Mb,Nb)) ).

% int_numeral_and_not_num
tff(fact_5194_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)) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(Nb,Mb)) ).

% int_numeral_not_and_num
tff(fact_5195_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_mc(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_5196_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_5197_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_md(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Max.eq_fold'
tff(fact_5198_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,aa(num,B,aa(fun(num,B),fun(num,B),aa(fun(num,B),fun(fun(num,B),fun(num,B)),aa(B,fun(fun(num,B),fun(fun(num,B),fun(num,B))),case_num(B),F1),F22),F32),Num)) = aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),aa(B,A,H,F1)),aa(fun(num,B),fun(num,A),aTP_Lamp_me(fun(B,A),fun(fun(num,B),fun(num,A)),H),F22)),aa(fun(num,B),fun(num,A),aTP_Lamp_me(fun(B,A),fun(fun(num,B),fun(num,A)),H),F32)),Num) ).

% num.case_distrib
tff(fact_5199_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X22: num] : aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),aa(num,num,bit0,X22)) = aa(num,A,F22,X22) ).

% verit_eq_simplify(17)
tff(fact_5200_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A)] : aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_5201_verit__eq__simplify_I18_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X32: num] : aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),aa(num,num,bit1,X32)) = aa(num,A,F32,X32) ).

% verit_eq_simplify(18)
tff(fact_5202_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) ) ) )
           => ( ! [M2: num] :
                  ( ( Xb = aa(num,num,bit0,M2) )
                 => ( ( Xaa = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M2)) ) ) )
             => ( ! [M2: num] :
                    ( ( Xb = aa(num,num,bit0,M2) )
                   => ! [N: num] :
                        ( ( Xaa = aa(num,num,bit0,N) )
                       => ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M2,N)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xb = aa(num,num,bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa(num,num,bit1,N) )
                         => ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M2,N)) ) ) )
                 => ( ! [M2: num] :
                        ( ( Xb = aa(num,num,bit1,M2) )
                       => ( ( Xaa = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M2)) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xb = aa(num,num,bit1,M2) )
                         => ! [N: num] :
                              ( ( Xaa = aa(num,num,bit0,N) )
                             => ( Y != aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_lu(num,option(num))),bit_and_not_num(M2,N)) ) ) )
                     => ~ ! [M2: num] :
                            ( ( Xb = aa(num,num,bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa(num,num,bit1,N) )
                               => ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M2,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_5203_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),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),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_5204_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),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),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_5205_map__option__eq__Some,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xo: option(B),Y: A] :
      ( ( aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),Xo) = aa(A,option(A),some(A),Y) )
    <=> ? [Z2: B] :
          ( ( Xo = aa(B,option(B),some(B),Z2) )
          & ( aa(B,A,F2,Z2) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_5206_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xb: option(B)] :
      ( ( none(A) = aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),Xb) )
    <=> ( Xb = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_5207_map__option__is__None,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Opt: option(B)] :
      ( ( aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),Opt) = none(A) )
    <=> ( Opt = none(B) ) ) ).

% map_option_is_None
tff(fact_5208_option_Omap__disc__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: option(B)] :
      ( ( aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),A3) = none(A) )
    <=> ( A3 = none(B) ) ) ).

% option.map_disc_iff
tff(fact_5209_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),aa(fun(num,num),fun(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_5210_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),aa(fun(num,num),fun(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_5211_option_Omap__ident,axiom,
    ! [A: $tType,Ta: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_mf(A,A)),Ta) = Ta ).

% option.map_ident
tff(fact_5212_option_Osimps_I8_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),none(B)) = none(A) ).

% option.simps(8)
tff(fact_5213_option_Osimps_I9_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X22: B] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),aa(B,option(B),some(B),X22)) = aa(A,option(A),some(A),aa(B,A,F2,X22)) ).

% option.simps(9)
tff(fact_5214_map__option__cong,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),Y: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xb = Y )
     => ( ! [A5: A] :
            ( ( Y = aa(A,option(A),some(A),A5) )
           => ( aa(A,B,F2,A5) = aa(A,B,G,A5) ) )
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),Xb) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Y) ) ) ) ).

% map_option_cong
tff(fact_5215_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),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(5)
tff(fact_5216_option_Omap__sel,axiom,
    ! [B: $tType,A: $tType,A3: option(A),F2: fun(A,B)] :
      ( ( A3 != none(A) )
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),A3)) = aa(A,B,F2,aa(option(A),A,the2(A),A3)) ) ) ).

% option.map_sel
tff(fact_5217_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),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(6)
tff(fact_5218_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),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(9)
tff(fact_5219_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_5220_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)) ) ) )
           => ( ! [M2: num] :
                  ( ( Xb = aa(num,num,bit0,M2) )
                 => ( ( Xaa = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit1,M2)) ) ) )
             => ( ! [M2: num] :
                    ( ( Xb = aa(num,num,bit0,M2) )
                   => ! [N: num] :
                        ( ( Xaa = aa(num,num,bit0,N) )
                       => ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M2),N)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xb = aa(num,num,bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa(num,num,bit1,N) )
                         => ( Y != aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M2),N))) ) ) )
                 => ( ! [M2: num] :
                        ( ( Xb = aa(num,num,bit1,M2) )
                       => ( ( Xaa = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M2)) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xb = aa(num,num,bit1,M2) )
                         => ! [N: num] :
                              ( ( Xaa = aa(num,num,bit0,N) )
                             => ( Y != aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M2),N))) ) ) )
                     => ~ ! [M2: num] :
                            ( ( Xb = aa(num,num,bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa(num,num,bit1,N) )
                               => ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M2),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_5221_map__option__case,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Y: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),Y) = aa(option(B),option(A),aa(fun(B,option(A)),fun(option(B),option(A)),aa(option(A),fun(fun(B,option(A)),fun(option(B),option(A))),case_option(option(A),B),none(A)),aTP_Lamp_mg(fun(B,A),fun(B,option(A)),F2)),Y) ).

% map_option_case
tff(fact_5222_xor__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num,Q2: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Mb),Nb) = aa(num,option(num),some(num),Q2) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% xor_num_eq_Some_iff
tff(fact_5223_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_5224_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_5225_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_5226_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_5227_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_5228_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)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),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_5229_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) ) ) )
           => ( ( ? [M2: num] : Xb = aa(num,num,bit0,M2)
               => ( ( Xaa = one2 )
                 => ( Y != none(num) ) ) )
             => ( ! [M2: num] :
                    ( ( Xb = aa(num,num,bit0,M2) )
                   => ! [N: num] :
                        ( ( Xaa = aa(num,num,bit0,N) )
                       => ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M2),N)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xb = aa(num,num,bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa(num,num,bit1,N) )
                         => ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M2),N)) ) ) )
                 => ( ( ? [M2: num] : Xb = aa(num,num,bit1,M2)
                     => ( ( Xaa = one2 )
                       => ( Y != aa(num,option(num),some(num),one2) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xb = aa(num,num,bit1,M2) )
                         => ! [N: num] :
                              ( ( Xaa = aa(num,num,bit0,N) )
                             => ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M2),N)) ) ) )
                     => ~ ! [M2: num] :
                            ( ( Xb = aa(num,num,bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa(num,num,bit1,N) )
                               => ( Y != aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_lu(num,option(num))),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M2),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_5230_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_5231_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_5232_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_5233_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_5234_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_5235_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_5236_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_5237_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_5238_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),aa(fun(num,num),fun(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_5239_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys))
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
             => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
         => ( take(A,K,Xs) = take(A,K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_5240_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_5241_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_5242_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_5243_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_5244_and__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num,Q2: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb) = aa(num,option(num),some(num),Q2) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% and_num_eq_Some_iff
tff(fact_5245_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),aa(fun(num,num),fun(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_5246_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),aa(fun(num,num),fun(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_5247_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_5248_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)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),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_5249_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),aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($o,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Bs)) = aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($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_5250_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)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_lu(num,option(num))),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(9)
tff(fact_5251_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_5252_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R))
     => ~ ! [I2: nat] :
            ( aa(nat,$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)),R) ) ) ) ) ).

% lex_take_index
tff(fact_5253_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_kw(A,fun(A,$o))) = ord_max(A) ) ) ).

% dual_min
tff(fact_5254_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R))
    <=> ? [Y3: A,N4: nat] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N4)),Y3),R)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
          & ( Ys = list_update(A,Xs,N4,Y3) ) ) ) ).

% listrel1_iff_update
tff(fact_5255_lenlex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_mh(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R))) ).

% lenlex_conv
tff(fact_5256_card__Min__le__sum,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( finite_finite(A,A4)
     => aa(nat,$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),A4)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),image(A,nat,F2,A4)))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A4)) ) ).

% card_Min_le_sum
tff(fact_5257_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A4))
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_5258_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A4))
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_5259_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( member(A,Xb,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),Xb) ) ) ) ).

% Min_le
tff(fact_5260_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ! [Y4: A] :
                ( member(A,Y4,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y4) )
           => ( member(A,Xb,A4)
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = Xb ) ) ) ) ) ).

% Min_eqI
tff(fact_5261_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite(A,A4)
         => ( member(A,A3,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),A3) ) ) ) ).

% Min.coboundedI
tff(fact_5262_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: 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,R))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_5263_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Mb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = Mb )
            <=> ( member(A,Mb,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_5264_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),Xb)
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xb) ) ) ) ) ) ).

% Min_le_iff
tff(fact_5265_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Mb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( Mb = aa(set(A),A,lattic643756798350308766er_Min(A),A4) )
            <=> ( member(A,Mb,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_5266_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A4))
             => ! [A9: A] :
                  ( member(A,A9,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A9) ) ) ) ) ) ).

% Min.boundedE
tff(fact_5267_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A5) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ).

% Min.boundedI
tff(fact_5268_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),Xb)
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xb) ) ) ) ) ) ).

% Min_less_iff
tff(fact_5269_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite(A,A4)
         => ( ! [B3: A] :
                ( member(A,B3,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),insert(A,A3),A4)) = A3 ) ) ) ) ).

% Min_insert2
tff(fact_5270_lenlex__irreflexive,axiom,
    ! [A: $tType,R: 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),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),Xs),lenlex(A,R)) ) ).

% lenlex_irreflexive
tff(fact_5271_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( ~ finite_finite(A,A4)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Min.infinite
tff(fact_5272_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( finite_finite(A,B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B4)),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ).

% Min.subset_imp
tff(fact_5273_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M6),N2)
         => ( ( M6 != bot_bot(set(A)) )
           => ( finite_finite(A,N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N2)),aa(set(A),A,lattic643756798350308766er_Min(A),M6)) ) ) ) ) ).

% Min_antimono
tff(fact_5274_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S3: set(A),F2: fun(A,B),K: B] :
          ( 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_ip(fun(A,B),fun(B,fun(A,B)),F2),K),S3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798350308766er_Min(B),image(A,B,F2,S3))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_5275_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)) ) ).

% lenlex_length
tff(fact_5276_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_kw(A,fun(A,$o))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_5277_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_mi(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Min.eq_fold'
tff(fact_5278_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_5279_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),Q2),zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_5280_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),zero_zero(extended_enat)),Q2) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_5281_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb1
tff(fact_5282_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_5283_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% min.bounded_iff
tff(fact_5284_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_5285_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_5286_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb3
tff(fact_5287_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_5288_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_5289_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_5290_take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)),A3) ) ).

% take_bit_take_bit
tff(fact_5291_signed__take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Mb),aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A3)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)),A3) ) ).

% signed_take_bit_signed_take_bit
tff(fact_5292_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_5293_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_5294_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_5295_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_5296_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_5297_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_5298_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_5299_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_5300_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_5301_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_5302_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_5303_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_5304_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_5305_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K)),Nb)) ).

% min_numeral_Suc
tff(fact_5306_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_5307_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_5308_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2),A3,B2) ) ).

% min_def
tff(fact_5309_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_5310_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).

% min.coboundedI2
tff(fact_5311_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).

% min.coboundedI1
tff(fact_5312_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_5313_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb_iff1
tff(fact_5314_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2) ) ).

% min.cobounded2
tff(fact_5315_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3) ) ).

% min.cobounded1
tff(fact_5316_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.order_iff
tff(fact_5317_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ) ) ).

% min.boundedI
tff(fact_5318_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% min.boundedE
tff(fact_5319_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% min.orderI
tff(fact_5320_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.orderE
tff(fact_5321_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D2)) ) ) ) ).

% min.mono
tff(fact_5322_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),X5),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Xa),X5,Xa) ) ).

% min_def_raw
tff(fact_5323_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).

% min.strict_coboundedI2
tff(fact_5324_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).

% min.strict_coboundedI1
tff(fact_5325_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_5326_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% min.strict_boundedE
tff(fact_5327_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_5328_nat__mult__min__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),Q2)) = 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),Q2)) ).

% nat_mult_min_right
tff(fact_5329_nat__mult__min__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ).

% nat_mult_min_left
tff(fact_5330_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_5331_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_5332_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_5333_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_5334_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_5335_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_5336_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_5337_concat__bit__assoc__sym,axiom,
    ! [Mb: nat,Nb: nat,K: int,L: int,R: int] : aa(int,int,bit_concat_bit(Mb,aa(int,int,bit_concat_bit(Nb,K),L)),R) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb),K),aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),L),R)) ).

% concat_bit_assoc_sym
tff(fact_5338_take__bit__concat__bit__eq,axiom,
    ! [Mb: nat,Nb: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),aa(int,int,bit_concat_bit(Nb,K),L)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb),K),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_5339_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_5340_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_5341_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_5342_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_5343_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,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),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2))) ) ).

% min_divide_distrib_right
tff(fact_5344_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,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),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2))) ) ).

% max_divide_distrib_right
tff(fact_5345_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_mj(nat,fun(nat,nat),Nb),Mb) ).

% min_Suc1
tff(fact_5346_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_mk(nat,fun(nat,nat),Nb),Mb) ).

% min_Suc2
tff(fact_5347_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Mb: nat,Nb: nat] : modulo_modulo(A,modulo_modulo(A,A3,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,A3,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_5348_lexord__take__index__conv,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Y),lexord(A,R))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xb)),aa(list(A),nat,size_size(list(A)),Y))
          & ( take(A,aa(list(A),nat,size_size(list(A)),Xb),Y) = Xb ) )
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),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,I4,Xb) = take(A,I4,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),I4)),aa(nat,A,nth(A,Y),I4)),R) ) ) ) ).

% lexord_take_index_conv
tff(fact_5349_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_kw(A,fun(A,$o))) = ord_min(A) ) ) ).

% dual_max
tff(fact_5350_card__Pow,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite(A,A4)
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow2(A,A4)) = 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),A4)) ) ) ).

% card_Pow
tff(fact_5351_lexord__irreflexive,axiom,
    ! [A: $tType,R: 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),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),Xs),lexord(A,R)) ) ).

% lexord_irreflexive
tff(fact_5352_lexord__linear,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xb: list(A),Y: list(A)] :
      ( ! [A5: A,B3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B3),R)
          | ( A5 = B3 )
          | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A5),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)),Xb),Y),lexord(A,R))
        | ( 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,R)) ) ) ).

% lexord_linear
tff(fact_5353_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X4: A,Y4: 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),Y4),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),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),X4),Z3),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),lexord(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)),Ys),Zs),lexord(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),Zs),lexord(A,R)) ) ) ) ).

% lexord_partial_trans
tff(fact_5354_lexord__lex,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R: 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,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)),Xb),Y),lexord(A,R))
        & ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_5355_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Mb: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K,Mb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_5356_quotient__of__def,axiom,
    ! [Xb: rat] : quotient_of(Xb) = the(product_prod(int,int),aTP_Lamp_ml(rat,fun(product_prod(int,int),$o),Xb)) ).

% quotient_of_def
tff(fact_5357_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_5358_coprime__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
        <=> ( algebr8660921524188924756oprime(A,C2,A3)
            & algebr8660921524188924756oprime(A,C2,B2) ) ) ) ).

% coprime_mult_right_iff
tff(fact_5359_coprime__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
        <=> ( algebr8660921524188924756oprime(A,A3,C2)
            & algebr8660921524188924756oprime(A,B2,C2) ) ) ) ).

% coprime_mult_left_iff
tff(fact_5360_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_5361_coprime__abs__right__iff,axiom,
    ! [K: int,L: int] :
      ( algebr8660921524188924756oprime(int,K,aa(int,int,abs_abs(int),L))
    <=> algebr8660921524188924756oprime(int,K,L) ) ).

% coprime_abs_right_iff
tff(fact_5362_coprime__abs__left__iff,axiom,
    ! [K: int,L: int] :
      ( algebr8660921524188924756oprime(int,aa(int,int,abs_abs(int),K),L)
    <=> algebr8660921524188924756oprime(int,K,L) ) ).

% coprime_abs_left_iff
tff(fact_5363_coprime__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,A3)
        <=> dvd_dvd(A,A3,one_one(A)) ) ) ).

% coprime_self
tff(fact_5364_coprime__power__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,Nb: nat,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb),B2)
        <=> ( algebr8660921524188924756oprime(A,A3,B2)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_5365_coprime__power__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,Nb: nat] :
          ( algebr8660921524188924756oprime(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
        <=> ( algebr8660921524188924756oprime(A,A3,B2)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_5366_coprime__mod__right__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,A3,modulo_modulo(A,B2,A3))
          <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mod_right_iff
tff(fact_5367_coprime__mod__left__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,modulo_modulo(A,A3,B2),B2)
          <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mod_left_iff
tff(fact_5368_coprime__imp__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).

% coprime_imp_gcd_eq_1
tff(fact_5369_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A3)
        <=> dvd_dvd(A,A3,one_one(A)) ) ) ).

% coprime_0_left_iff
tff(fact_5370_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,zero_zero(A))
        <=> dvd_dvd(A,A3,one_one(A)) ) ) ).

% coprime_0_right_iff
tff(fact_5371_coprime__mult__self__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( dvd_dvd(A,C2,one_one(A))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_5372_coprime__mult__self__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( dvd_dvd(A,C2,one_one(A))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_5373_is__unit__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2),one_one(A))
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_gcd
tff(fact_5374_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_5375_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_5376_coprime__left__2__iff__odd,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ).

% coprime_left_2_iff_odd
tff(fact_5377_coprime__right__2__iff__odd,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,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)),A3) ) ) ).

% coprime_right_2_iff_odd
tff(fact_5378_normalize__stable,axiom,
    ! [Q2: int,P2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2)
     => ( algebr8660921524188924756oprime(int,P2,Q2)
       => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) ) ) ) ).

% normalize_stable
tff(fact_5379_coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ! [C3: A] :
              ( dvd_dvd(A,C3,A3)
             => ( dvd_dvd(A,C3,B2)
               => dvd_dvd(A,C3,one_one(A)) ) )
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprimeI
tff(fact_5380_coprime__def,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
        <=> ! [C4: A] :
              ( dvd_dvd(A,C4,A3)
             => ( dvd_dvd(A,C4,B2)
               => dvd_dvd(A,C4,one_one(A)) ) ) ) ) ).

% coprime_def
tff(fact_5381_not__coprimeE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ algebr8660921524188924756oprime(A,A3,B2)
         => ~ ! [C3: A] :
                ( dvd_dvd(A,C3,A3)
               => ( dvd_dvd(A,C3,B2)
                 => dvd_dvd(A,C3,one_one(A)) ) ) ) ) ).

% not_coprimeE
tff(fact_5382_not__coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( dvd_dvd(A,C2,A3)
         => ( dvd_dvd(A,C2,B2)
           => ( ~ dvd_dvd(A,C2,one_one(A))
             => ~ algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% not_coprimeI
tff(fact_5383_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_5384_coprime__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,D2: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,D2)
         => ( ! [E2: A] :
                ( ~ dvd_dvd(A,E2,one_one(A))
               => ( dvd_dvd(A,E2,A3)
                 => ( dvd_dvd(A,E2,B2)
                   => dvd_dvd(A,E2,C2) ) ) )
           => ( ! [E2: A] :
                  ( ~ dvd_dvd(A,E2,one_one(A))
                 => ( dvd_dvd(A,E2,A3)
                   => ( dvd_dvd(A,E2,B2)
                     => dvd_dvd(A,E2,D2) ) ) )
             => algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_5385_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_5386_coprime__common__divisor,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( dvd_dvd(A,C2,A3)
           => ( dvd_dvd(A,C2,B2)
             => dvd_dvd(A,C2,one_one(A)) ) ) ) ) ).

% coprime_common_divisor
tff(fact_5387_is__unit__left__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_left_imp_coprime
tff(fact_5388_is__unit__right__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_right_imp_coprime
tff(fact_5389_mult__mod__cancel__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [A3: A,Nb: A,Mb: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),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,A3,Mb) = modulo_modulo(A,B2,Mb) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_5390_mult__mod__cancel__left,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [Nb: A,A3: A,Mb: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),Nb),A3),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,A3,Mb) = modulo_modulo(A,B2,Mb) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_5391_coprime__add__one__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).

% coprime_add_one_right
tff(fact_5392_coprime__add__one__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),A3) ) ).

% coprime_add_one_left
tff(fact_5393_coprime__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,A3)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_commute
tff(fact_5394_coprime__divisors,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( dvd_dvd(A,A3,C2)
         => ( dvd_dvd(A,B2,D2)
           => ( algebr8660921524188924756oprime(A,C2,D2)
             => algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% coprime_divisors
tff(fact_5395_coprime__1__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).

% coprime_1_right
tff(fact_5396_coprime__1__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).

% coprime_1_left
tff(fact_5397_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_5398_coprime__doff__one__right,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).

% coprime_doff_one_right
tff(fact_5399_coprime__diff__one__left,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A)),A3) ) ).

% coprime_diff_one_left
tff(fact_5400_divides__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( dvd_dvd(A,A3,C2)
         => ( dvd_dvd(A,B2,C2)
           => ( algebr8660921524188924756oprime(A,A3,B2)
             => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ) ) ) ).

% divides_mult
tff(fact_5401_coprime__dvd__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> dvd_dvd(A,A3,B2) ) ) ) ).

% coprime_dvd_mult_left_iff
tff(fact_5402_coprime__dvd__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> dvd_dvd(A,A3,B2) ) ) ) ).

% coprime_dvd_mult_right_iff
tff(fact_5403_gcd__mult__right__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_right_right_cancel
tff(fact_5404_gcd__mult__right__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_right_left_cancel
tff(fact_5405_gcd__mult__left__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_left_right_cancel
tff(fact_5406_gcd__mult__left__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_left_left_cancel
tff(fact_5407_gcd__eq__1__imp__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% gcd_eq_1_imp_coprime
tff(fact_5408_coprime__iff__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).

% coprime_iff_gcd_eq_1
tff(fact_5409_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_5410_invertible__coprime,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A3,C2) ) ) ).

% invertible_coprime
tff(fact_5411_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
         => ? [A10: A,B8: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A10),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B8),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & algebr8660921524188924756oprime(A,A10,B8) ) ) ) ).

% gcd_coprime_exists
tff(fact_5412_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,A7: A,B6: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A7),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
           => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
             => algebr8660921524188924756oprime(A,A7,B6) ) ) ) ) ).

% gcd_coprime
tff(fact_5413_div__gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( ( A3 != zero_zero(A) )
            | ( B2 != zero_zero(A) ) )
         => algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ) ).

% div_gcd_coprime
tff(fact_5414_Rat__induct,axiom,
    ! [P: fun(rat,$o),Q2: rat] :
      ( ! [A5: int,B3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
         => ( algebr8660921524188924756oprime(int,A5,B3)
           => aa(rat,$o,P,fract(A5,B3)) ) )
     => aa(rat,$o,P,Q2) ) ).

% Rat_induct
tff(fact_5415_Rat__cases,axiom,
    ! [Q2: rat] :
      ~ ! [A5: int,B3: int] :
          ( ( Q2 = fract(A5,B3) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
           => ~ algebr8660921524188924756oprime(int,A5,B3) ) ) ).

% Rat_cases
tff(fact_5416_coprime__common__divisor__int,axiom,
    ! [A3: int,B2: int,Xb: int] :
      ( algebr8660921524188924756oprime(int,A3,B2)
     => ( dvd_dvd(int,Xb,A3)
       => ( dvd_dvd(int,Xb,B2)
         => ( aa(int,int,abs_abs(int),Xb) = one_one(int) ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_5417_le__prod__encode__1,axiom,
    ! [A3: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),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),A3),B2))) ).

% le_prod_encode_1
tff(fact_5418_le__prod__encode__2,axiom,
    ! [B2: nat,A3: 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),A3),B2))) ).

% le_prod_encode_2
tff(fact_5419_Rat__cases__nonzero,axiom,
    ! [Q2: rat] :
      ( ! [A5: int,B3: int] :
          ( ( Q2 = fract(A5,B3) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
           => ( ( A5 != zero_zero(int) )
             => ~ algebr8660921524188924756oprime(int,A5,B3) ) ) )
     => ( Q2 = zero_zero(rat) ) ) ).

% Rat_cases_nonzero
tff(fact_5420_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_mm(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_5421_quotient__of__unique,axiom,
    ! [R: rat] :
    ? [X4: product_prod(int,int)] :
      ( ( R = fract(aa(product_prod(int,int),int,product_fst(int,int),X4),aa(product_prod(int,int),int,product_snd(int,int),X4)) )
      & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),X4))
      & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),X4),aa(product_prod(int,int),int,product_snd(int,int),X4))
      & ! [Y5: product_prod(int,int)] :
          ( ( ( R = fract(aa(product_prod(int,int),int,product_fst(int,int),Y5),aa(product_prod(int,int),int,product_snd(int,int),Y5)) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Y5))
            & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y5),aa(product_prod(int,int),int,product_snd(int,int),Y5)) )
         => ( Y5 = X4 ) ) ) ).

% quotient_of_unique
tff(fact_5422_Rats__cases_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xb: A] :
          ( member(A,Xb,field_char_0_Rats(A))
         => ~ ! [A5: int,B3: int] :
                ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
               => ( algebr8660921524188924756oprime(int,A5,B3)
                 => ( Xb != aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A5)),aa(int,A,ring_1_of_int(A),B3)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_5423_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_5424_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) )
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),I4))
          & ( Xb = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_5425_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_5426_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_5427_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_5428_coprime__nat__abs__left__iff,axiom,
    ! [K: int,Nb: nat] :
      ( algebr8660921524188924756oprime(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),Nb)
    <=> algebr8660921524188924756oprime(int,K,aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% coprime_nat_abs_left_iff
tff(fact_5429_coprime__nat__abs__right__iff,axiom,
    ! [Nb: nat,K: int] :
      ( algebr8660921524188924756oprime(nat,Nb,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
    <=> algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),Nb),K) ) ).

% coprime_nat_abs_right_iff
tff(fact_5430_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_5431_coprime__common__divisor__nat,axiom,
    ! [A3: nat,B2: nat,Xb: nat] :
      ( algebr8660921524188924756oprime(nat,A3,B2)
     => ( dvd_dvd(nat,Xb,A3)
       => ( dvd_dvd(nat,Xb,B2)
         => ( Xb = one_one(nat) ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_5432_coprime__crossproduct__nat,axiom,
    ! [A3: nat,D2: nat,B2: nat,C2: nat] :
      ( algebr8660921524188924756oprime(nat,A3,D2)
     => ( algebr8660921524188924756oprime(nat,B2,C2)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),C2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),D2) )
        <=> ( ( A3 = B2 )
            & ( C2 = D2 ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_5433_coprime__Suc__0__left,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) ).

% coprime_Suc_0_left
tff(fact_5434_coprime__Suc__0__right,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,Nb,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_5435_Rats__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,field_char_0_Rats(A))
         => ( member(A,B2,field_char_0_Rats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_divide
tff(fact_5436_Rats__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => member(A,one_one(A),field_char_0_Rats(A)) ) ).

% Rats_1
tff(fact_5437_Rats__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,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),A3),B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_add
tff(fact_5438_Rats__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,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),A3),B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_mult
tff(fact_5439_Rats__power,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Nb: nat] :
          ( member(A,A3,field_char_0_Rats(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),Nb),field_char_0_Rats(A)) ) ) ).

% Rats_power
tff(fact_5440_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_5441_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_5442_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_5443_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_5444_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_5445_Rats__abs__nat__div__natE,axiom,
    ! [Xb: real] :
      ( member(real,Xb,field_char_0_Rats(real))
     => ~ ! [M2: nat,N: nat] :
            ( ( N != zero_zero(nat) )
           => ( ( aa(real,real,abs_abs(real),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),M2)),aa(nat,real,semiring_1_of_nat(real),N)) )
             => ~ algebr8660921524188924756oprime(nat,M2,N) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_5446_zip__same,axiom,
    ! [A: $tType,A3: 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),A3),B2),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs)))
    <=> ( member(A,A3,aa(list(A),set(A),set2(A),Xs))
        & ( A3 = B2 ) ) ) ).

% zip_same
tff(fact_5447_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_5448_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_5449_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_5450_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_5451_find__None__iff2,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( none(A) = find(A,P,Xs) )
    <=> ~ ? [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X) ) ) ).

% find_None_iff2
tff(fact_5452_find__None__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( find(A,P,Xs) = none(A) )
    <=> ~ ? [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X) ) ) ).

% find_None_iff
tff(fact_5453_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_5454_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_5455_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))
       => ~ ! [Y4: 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),Y4),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_5456_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_5457_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_5458_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) )
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),I4))
          & ( Xb = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_5459_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),$o),aTP_Lamp_mn(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys)) ).

% set_zip
tff(fact_5460_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
         => ( aa(list(A),A,groups8242544230860333062m_list(A),list_update(A,Xs,K,Xb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),Xb)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_5461_and__not__num_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: option(num)] :
      ( ( bit_and_not_num(Xb,Xaa) = Y )
     => ( accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),Xb),Xaa))
       => ( ( ( Xb = one2 )
           => ( ( Xaa = one2 )
             => ( ( Y = none(num) )
               => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( Xb = one2 )
             => ! [N: num] :
                  ( ( Xaa = aa(num,num,bit0,N) )
                 => ( ( Y = aa(num,option(num),some(num),one2) )
                   => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N))) ) ) )
           => ( ( ( Xb = one2 )
               => ! [N: num] :
                    ( ( Xaa = aa(num,num,bit1,N) )
                   => ( ( Y = none(num) )
                     => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N))) ) ) )
             => ( ! [M2: num] :
                    ( ( Xb = aa(num,num,bit0,M2) )
                   => ( ( Xaa = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M2)) )
                       => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),one2)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xb = aa(num,num,bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa(num,num,bit0,N) )
                         => ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M2,N)) )
                           => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),aa(num,num,bit0,N))) ) ) )
                 => ( ! [M2: num] :
                        ( ( Xb = aa(num,num,bit0,M2) )
                       => ! [N: num] :
                            ( ( Xaa = aa(num,num,bit1,N) )
                           => ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M2,N)) )
                             => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),aa(num,num,bit1,N))) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xb = aa(num,num,bit1,M2) )
                         => ( ( Xaa = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M2)) )
                             => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),one2)) ) ) )
                     => ( ! [M2: num] :
                            ( ( Xb = aa(num,num,bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa(num,num,bit0,N) )
                               => ( ( Y = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_lu(num,option(num))),bit_and_not_num(M2,N)) )
                                 => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),aa(num,num,bit0,N))) ) ) )
                       => ~ ! [M2: num] :
                              ( ( Xb = aa(num,num,bit1,M2) )
                             => ! [N: num] :
                                  ( ( Xaa = aa(num,num,bit1,N) )
                                 => ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M2,N)) )
                                   => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),aa(num,num,bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_5462_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),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ) ).

% member_le_sum_list
tff(fact_5463_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)),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_5464_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) )
         => ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
          <=> ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
               => ( X = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_5465_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),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_5466_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat,C2: A] : aa(list(A),A,groups8242544230860333062m_list(A),replicate(A,Nb,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),C2) ) ).

% sum_list_replicate
tff(fact_5467_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),aa(list(A),A,groups8242544230860333062m_list(A),Ns)) ) ) ).

% elem_le_sum_list
tff(fact_5468_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_mo(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_5469_card__length__sum__list__rec,axiom,
    ! [Mb: nat,N2: 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_mp(nat,fun(nat,fun(list(nat),$o)),Mb),N2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_mq(nat,fun(nat,fun(list(nat),$o)),Mb),N2)))),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_mr(nat,fun(nat,fun(list(nat),$o)),Mb),N2)))) ) ) ).

% card_length_sum_list_rec
tff(fact_5470_card__length__sum__list,axiom,
    ! [Mb: nat,N2: 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_mp(nat,fun(nat,fun(list(nat),$o)),Mb),N2))) = 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),N2),Mb)),one_one(nat))),N2) ).

% card_length_sum_list
tff(fact_5471_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] : aa(list(A),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_5472_and__num_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Xb),Xaa) = Y )
     => ( accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),Xb),Xaa))
       => ( ( ( Xb = one2 )
           => ( ( Xaa = one2 )
             => ( ( Y = aa(num,option(num),some(num),one2) )
               => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( Xb = one2 )
             => ! [N: num] :
                  ( ( Xaa = aa(num,num,bit0,N) )
                 => ( ( Y = none(num) )
                   => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N))) ) ) )
           => ( ( ( Xb = one2 )
               => ! [N: num] :
                    ( ( Xaa = aa(num,num,bit1,N) )
                   => ( ( Y = aa(num,option(num),some(num),one2) )
                     => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N))) ) ) )
             => ( ! [M2: num] :
                    ( ( Xb = aa(num,num,bit0,M2) )
                   => ( ( Xaa = one2 )
                     => ( ( Y = none(num) )
                       => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),one2)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xb = aa(num,num,bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa(num,num,bit0,N) )
                         => ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M2),N)) )
                           => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),aa(num,num,bit0,N))) ) ) )
                 => ( ! [M2: num] :
                        ( ( Xb = aa(num,num,bit0,M2) )
                       => ! [N: num] :
                            ( ( Xaa = aa(num,num,bit1,N) )
                           => ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M2),N)) )
                             => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),aa(num,num,bit1,N))) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xb = aa(num,num,bit1,M2) )
                         => ( ( Xaa = one2 )
                           => ( ( Y = aa(num,option(num),some(num),one2) )
                             => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),one2)) ) ) )
                     => ( ! [M2: num] :
                            ( ( Xb = aa(num,num,bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa(num,num,bit0,N) )
                               => ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M2),N)) )
                                 => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),aa(num,num,bit0,N))) ) ) )
                       => ~ ! [M2: num] :
                              ( ( Xb = aa(num,num,bit1,M2) )
                             => ! [N: num] :
                                  ( ( Xaa = aa(num,num,bit1,N) )
                                 => ( ( Y = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_lu(num,option(num))),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M2),N)) )
                                   => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),aa(num,num,bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_5473_xor__num_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Xb),Xaa) = Y )
     => ( accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),Xb),Xaa))
       => ( ( ( Xb = one2 )
           => ( ( Xaa = one2 )
             => ( ( Y = none(num) )
               => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( Xb = one2 )
             => ! [N: num] :
                  ( ( Xaa = aa(num,num,bit0,N) )
                 => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,N)) )
                   => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N))) ) ) )
           => ( ( ( Xb = one2 )
               => ! [N: num] :
                    ( ( Xaa = aa(num,num,bit1,N) )
                   => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,N)) )
                     => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N))) ) ) )
             => ( ! [M2: num] :
                    ( ( Xb = aa(num,num,bit0,M2) )
                   => ( ( Xaa = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,M2)) )
                       => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),one2)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xb = aa(num,num,bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa(num,num,bit0,N) )
                         => ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M2),N)) )
                           => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),aa(num,num,bit0,N))) ) ) )
                 => ( ! [M2: num] :
                        ( ( Xb = aa(num,num,bit0,M2) )
                       => ! [N: num] :
                            ( ( Xaa = aa(num,num,bit1,N) )
                           => ( ( Y = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M2),N))) )
                             => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M2)),aa(num,num,bit1,N))) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xb = aa(num,num,bit1,M2) )
                         => ( ( Xaa = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M2)) )
                             => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),one2)) ) ) )
                     => ( ! [M2: num] :
                            ( ( Xb = aa(num,num,bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa(num,num,bit0,N) )
                               => ( ( Y = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M2),N))) )
                                 => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),aa(num,num,bit0,N))) ) ) )
                       => ~ ! [M2: num] :
                              ( ( Xb = aa(num,num,bit1,M2) )
                             => ! [N: num] :
                                  ( ( Xaa = aa(num,num,bit1,N) )
                                 => ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M2),N)) )
                                   => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),aa(num,num,bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_5474_or__not__num__neg_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( bit_or_not_num_neg(Xb,Xaa) = Y )
     => ( accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),Xb),Xaa))
       => ( ( ( Xb = one2 )
           => ( ( Xaa = one2 )
             => ( ( Y = one2 )
               => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( Xb = one2 )
             => ! [M2: num] :
                  ( ( Xaa = aa(num,num,bit0,M2) )
                 => ( ( Y = aa(num,num,bit1,M2) )
                   => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,M2))) ) ) )
           => ( ( ( Xb = one2 )
               => ! [M2: num] :
                    ( ( Xaa = aa(num,num,bit1,M2) )
                   => ( ( Y = aa(num,num,bit1,M2) )
                     => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,M2))) ) ) )
             => ( ! [N: num] :
                    ( ( Xb = aa(num,num,bit0,N) )
                   => ( ( Xaa = one2 )
                     => ( ( Y = aa(num,num,bit0,one2) )
                       => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N)),one2)) ) ) )
               => ( ! [N: num] :
                      ( ( Xb = aa(num,num,bit0,N) )
                     => ! [M2: num] :
                          ( ( Xaa = aa(num,num,bit0,M2) )
                         => ( ( Y = bitM(bit_or_not_num_neg(N,M2)) )
                           => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N)),aa(num,num,bit0,M2))) ) ) )
                 => ( ! [N: num] :
                        ( ( Xb = aa(num,num,bit0,N) )
                       => ! [M2: num] :
                            ( ( Xaa = aa(num,num,bit1,M2) )
                           => ( ( Y = aa(num,num,bit0,bit_or_not_num_neg(N,M2)) )
                             => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N)),aa(num,num,bit1,M2))) ) ) )
                   => ( ! [N: num] :
                          ( ( Xb = aa(num,num,bit1,N) )
                         => ( ( Xaa = one2 )
                           => ( ( Y = one2 )
                             => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N)),one2)) ) ) )
                     => ( ! [N: num] :
                            ( ( Xb = aa(num,num,bit1,N) )
                           => ! [M2: num] :
                                ( ( Xaa = aa(num,num,bit0,M2) )
                               => ( ( Y = bitM(bit_or_not_num_neg(N,M2)) )
                                 => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N)),aa(num,num,bit0,M2))) ) ) )
                       => ~ ! [N: num] :
                              ( ( Xb = aa(num,num,bit1,N) )
                             => ! [M2: num] :
                                  ( ( Xaa = aa(num,num,bit1,M2) )
                                 => ( ( Y = bitM(bit_or_not_num_neg(N,M2)) )
                                   => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N)),aa(num,num,bit1,M2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_5475_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
tff(fact_5476_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
tff(fact_5477_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                          & aa(option(product_prod(nat,nat)),$o,
                              aa(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o),
                                aa($o,fun(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o)),case_option($o,product_prod(nat,nat)),
                                  ( ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(Summary,X7)
                                  & ! [X: vEBT_VEBT] :
                                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                     => ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(X,X7) ) )),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ms(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2))),
                              Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5478_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Deg3: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n: nat,
            n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
            $let(
              m2: nat,
              m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Dega),n),
              ( ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                 => vEBT_VEBT_valid(X,n) )
              & vEBT_VEBT_valid(Summarya,m2)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
              & aa(option(product_prod(nat,nat)),$o,
                  aa(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o),
                    aa($o,fun(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o)),case_option($o,product_prod(nat,nat)),
                      ( ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(Summarya,X7)
                      & ! [X: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                         => ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(X,X7) ) )),
                    aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ms(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Dega),TreeLista),Summarya),n),m2))),
                  Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5479_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                    $let(
                      m2: nat,
                      m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                      ( ! [X: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_VEBT_valid(X,n) )
                      & vEBT_VEBT_valid(Summary,m2)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                      & aa(option(product_prod(nat,nat)),$o,
                          aa(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o),
                            aa($o,fun(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o)),case_option($o,product_prod(nat,nat)),
                              ( ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(Summary,X7)
                              & ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(X,X7) ) )),
                            aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ms(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2))),
                          Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5480_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                      $let(
                        m2: nat,
                        m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                        ( ! [X: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_VEBT_valid(X,n) )
                        & vEBT_VEBT_valid(Summary,m2)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                        & aa(option(product_prod(nat,nat)),$o,
                            aa(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o),
                              aa($o,fun(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o)),case_option($o,product_prod(nat,nat)),
                                ( ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(Summary,X7)
                                & ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(X,X7) ) )),
                              aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ms(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2))),
                            Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_5481_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                            & aa(option(product_prod(nat,nat)),$o,
                                aa(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o),
                                  aa($o,fun(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o)),case_option($o,product_prod(nat,nat)),
                                    ( ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(Summary,X7)
                                    & ! [X: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                       => ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(X,X7) ) )),
                                  aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ms(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$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_5482_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                            & aa(option(product_prod(nat,nat)),$o,
                                aa(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o),
                                  aa($o,fun(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o)),case_option($o,product_prod(nat,nat)),
                                    ( ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(Summary,X7)
                                    & ! [X: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                       => ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(X,X7) ) )),
                                  aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ms(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2))),
                                Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_5483_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                          & aa(option(product_prod(nat,nat)),$o,
                              aa(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o),
                                aa($o,fun(fun(product_prod(nat,nat),$o),fun(option(product_prod(nat,nat)),$o)),case_option($o,product_prod(nat,nat)),
                                  ( ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(Summary,X7)
                                  & ! [X: vEBT_VEBT] :
                                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                     => ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(X,X7) ) )),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ms(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2))),
                              Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_5484_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_mt(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_5485_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6)
     => ( finite_finite(A,X6)
       => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,Xs)) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_mu(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F2)),X6) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_5486_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A4: set(C),F2: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A4,F2,G) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_mv(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),A4),F2),G)) ).

% image2_def
tff(fact_5487_length__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),map(B,A,F2,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_5488_nth__map,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),F2: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,map(A,B,F2,Xs)),Nb) = aa(A,B,F2,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% nth_map
tff(fact_5489_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(B,A),Xs: list(B),G: fun(C,A),Ys: list(C)] :
      ( ( map(B,A,F2,Xs) = map(C,A,G,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_5490_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : map(product_prod(B,C),A,F2,zip(B,C,Xs,map(D,C,G,Ys))) = 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_mw(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F2),G)),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_5491_zip__map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,F2: fun(C,A),Xs: list(C),G: fun(D,B),Ys: list(D)] : zip(A,B,map(C,A,F2,Xs),map(D,B,G,Ys)) = 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_mx(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F2),G)),zip(C,D,Xs,Ys)) ).

% zip_map_map
tff(fact_5492_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F2: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : map(product_prod(B,C),A,F2,zip(B,C,map(D,B,G,Xs),Ys)) = 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_my(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F2),G)),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_5493_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F2: fun(C,B),Ys: list(C)] : zip(A,B,Xs,map(C,B,F2,Ys)) = 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_mz(fun(C,B),fun(A,fun(C,product_prod(A,B))),F2)),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_5494_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,map(C,A,F2,Xs),Ys) = 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_na(fun(C,A),fun(C,fun(B,product_prod(A,B))),F2)),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_5495_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),C2: A,Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_bk(fun(B,A),fun(A,fun(B,A)),F2),C2),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F2,Xs))),C2) ) ).

% sum_list_mult_const
tff(fact_5496_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F2: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bj(A,fun(fun(B,A),fun(B,A)),C2),F2),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F2,Xs))) ) ).

% sum_list_const_mult
tff(fact_5497_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bn(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F2,Xs))),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,G,Xs))) ) ).

% sum_list_addf
tff(fact_5498_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : map(B,A,aTP_Lamp_nb(A,fun(B,A),K),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_5499_GreatestI__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y4: nat] :
            ( aa(nat,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_nat
tff(fact_5500_Greatest__le__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y4: nat] :
            ( aa(nat,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),order_Greatest(nat,P)) ) ) ).

% Greatest_le_nat
tff(fact_5501_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),B2: nat] :
      ( ? [X_13: nat] : aa(nat,$o,P,X_13)
     => ( ! [Y4: nat] :
            ( aa(nat,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_ex_nat
tff(fact_5502_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),aa(list(A),A,groups8242544230860333062m_list(A),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),map(A,A,abs_abs(A),Xs))) ) ).

% sum_list_abs
tff(fact_5503_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F2,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs))) ) ) ).

% sum_list_mono
tff(fact_5504_length__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,Xss)) = aa(list(nat),nat,groups8242544230860333062m_list(nat),map(list(A),nat,size_size(list(A)),Xss)) ).

% length_concat
tff(fact_5505_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
           => ( order_Greatest(A,P) = Xb ) ) ) ) ).

% Greatest_equality
tff(fact_5506_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
           => ( ! [X4: A] :
                  ( aa(A,$o,P,X4)
                 => ( ! [Y5: A] :
                        ( aa(A,$o,P,Y5)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),X4) )
                   => aa(A,$o,Q,X4) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_5507_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F2: fun(B,A),Xb: B,C2: C,G: fun(B,C),A4: set(B)] :
      ( ( B2 = aa(B,A,F2,Xb) )
     => ( ( C2 = aa(B,C,G,Xb) )
       => ( member(B,Xb,A4)
         => member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C2),bNF_Greatest_image2(B,A,C,A4,F2,G)) ) ) ) ).

% image2_eqI
tff(fact_5508_sum__list__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [R: A,Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aTP_Lamp_jt(A,fun(B,A),R),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))),R) ) ).

% sum_list_triv
tff(fact_5509_sum__list__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,aTP_Lamp_jx(fun(A,nat),fun(A,nat),F2),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_5510_Rats__eq__int__div__int,axiom,
    field_char_0_Rats(real) = aa(fun(real,$o),set(real),collect(real),aTP_Lamp_nc(real,$o)) ).

% Rats_eq_int_div_int
tff(fact_5511_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,Xs)) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_nd(fun(A,nat),fun(list(A),fun(A,nat)),F2),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_5512_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = aa(fun(real,$o),set(real),collect(real),aTP_Lamp_ne(real,$o)) ).

% Rats_eq_int_div_nat
tff(fact_5513_length__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = foldr(nat,nat,times_times(nat),map(list(A),nat,size_size(list(A)),Xss),one_one(nat)) ).

% length_product_lists
tff(fact_5514_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I5)) = aa(fun(A,$o),set(A),collect(A),aa(set(nat),fun(A,$o),aTP_Lamp_nf(list(A),fun(set(nat),fun(A,$o)),Xs),I5)) ).

% set_nths
tff(fact_5515_size__list__conv__sum__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : size_list(A,F2,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% size_list_conv_sum_list
tff(fact_5516_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) )
     => ( map(product_prod(A,B),A,product_fst(A,B),zip(A,B,Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_5517_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) )
     => ( map(product_prod(A,B),B,product_snd(A,B),zip(A,B,Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_5518_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_ng(list(B),fun(A,list(product_prod(A,B))),Ys),Xs)) ).

% product_concat_map
tff(fact_5519_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = map(A,product_prod(A,A),aTP_Lamp_nh(A,product_prod(A,A)),Xs) ).

% zip_same_conv_map
tff(fact_5520_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => member(nat,I2,I5) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_5521_size__list__estimation,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(A,nat,F2,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),size_list(A,F2,Xs)) ) ) ).

% size_list_estimation
tff(fact_5522_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X4)),aa(A,nat,G,X4)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),size_list(A,F2,Xs)),size_list(A,G,Xs)) ) ).

% size_list_pointwise
tff(fact_5523_size__list__estimation_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F2,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),size_list(A,F2,Xs)) ) ) ).

% size_list_estimation'
tff(fact_5524_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)) = 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_nj(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_5525_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)) = 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_nk(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_5526_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = 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_nl(B,fun(A,product_prod(A,B)))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_5527_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 )
      <=> ( ( map(product_prod(A,B),A,product_fst(A,B),Zs) = Xs )
          & ( map(product_prod(A,B),B,product_snd(A,B),Zs) = Ys ) ) ) ) ).

% zip_eq_conv
tff(fact_5528_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xb: A,Ys: list(B)] : zip(A,B,replicate(A,Nb,Xb),Ys) = 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_5529_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = foldr(A,A,plus_plus(A),Xs,zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_5530_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Nb: nat,Y: B] : zip(A,B,Xs,replicate(B,Nb,Y)) = map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_nl(B,fun(A,product_prod(A,B))),Y),take(A,Nb,Xs)) ).

% zip_replicate2
tff(fact_5531_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I5)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_nm(list(A),fun(set(nat),fun(nat,$o)),Xs),I5))) ).

% length_nths
tff(fact_5532_map__fst__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : 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_5533_map__snd__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(B),Ys: 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_5534_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_nn(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A3),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_5535_VEBT_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,size_size(vEBT_VEBT),X13)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size(3)
tff(fact_5536_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K: A,V1: B,V22: B] :
      ( distinct(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),K),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),K),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_5537_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_ng(list(B),fun(A,list(product_prod(A,B))),Ys),Xs))) ).

% product_code
tff(fact_5538_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,vEBT_size_VEBT,X13)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size_gen(1)
tff(fact_5539_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Nb: nat] : aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($o,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),map(nat,$o,bit_se5641148757651400278ts_bit(A,A3),upt(zero_zero(nat),Nb))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A3) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_5540_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Xb: A,Y: B] :
      ( distinct(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_5541_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,Xb: A] :
      ( distinct(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_5542_nth__upt,axiom,
    ! [I: nat,K: 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),K)),J)
     => ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).

% nth_upt
tff(fact_5543_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_5544_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_5545_map__fst__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : 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_5546_sum__list__upt,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(Mb,Nb)) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ce(nat,nat)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ) ).

% sum_list_upt
tff(fact_5547_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),Xb: A,Y: B] :
      ( distinct(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_5548_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : map(nat,nat,aTP_Lamp_no(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_5549_enumerate__map__upt,axiom,
    ! [A: $tType,Nb: nat,F2: fun(nat,A),Mb: nat] : enumerate(A,Nb,map(nat,A,F2,upt(Nb,Mb))) = map(nat,product_prod(nat,A),aTP_Lamp_np(fun(nat,A),fun(nat,product_prod(nat,A)),F2),upt(Nb,Mb)) ).

% enumerate_map_upt
tff(fact_5550_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_5551_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K: 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),K),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),K) = aa(B,option(B),some(B),X4) ) ).

% weak_map_of_SomeI
tff(fact_5552_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = 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),K),Y),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)) ) ).

% map_of_SomeD
tff(fact_5553_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : map(nat,A,nth(A,Xs),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_5554_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_5555_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Nb: nat,Mb: nat,F2: fun(nat,A)] :
      ( aa(nat,$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,map(nat,A,F2,upt(Mb,Nb))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I)) ) ) ).

% nth_map_upt
tff(fact_5556_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))
      <=> ? [Y3: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),Xb) = aa(B,option(B),some(B),Y3) ) ) ).

% map_of_zip_is_Some
tff(fact_5557_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_5558_enumerate__replicate__eq,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,A3: A] : enumerate(A,Nb,replicate(A,Mb,A3)) = map(nat,product_prod(nat,A),aTP_Lamp_nq(A,fun(nat,product_prod(nat,A)),A3),upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb))) ).

% enumerate_replicate_eq
tff(fact_5559_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Mb: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb) )
     => ( ! [I2: nat] :
            ( aa(nat,$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,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I2)) ) )
       => ( map(nat,A,F2,upt(Mb,Nb)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_5560_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F2: fun(A,B),X5: A] :
      aa(A,option(B),map_of(A,B,zip(A,B,Xs,map(A,B,F2,Xs))),X5) = $ite(member(A,X5,aa(list(A),set(A),set2(A),Xs)),aa(B,option(B),some(B),aa(A,B,F2,X5)),none(B)) ).

% map_of_zip_map
tff(fact_5561_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_5562_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xs))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),$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,$o),aTP_Lamp_nr(list(product_prod(A,B)),fun(A,fun(B,$o)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_5563_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F2: fun(nat,A),Ns: list(nat)] :
          ( ! [X4: nat,Y4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Y4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,X4)),aa(nat,A,F2,Y4)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(A),A,groups8242544230860333062m_list(A),map(nat,A,F2,Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_5564_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_5565_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A4: set(nat)] : nths(A,Xs,A4) = map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_ns(set(nat),fun(product_prod(A,nat),$o),A4),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_5566_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_nt(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_5567_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F2,filter2(B,P,Xs))) ) ) ).

% sorted_filter
tff(fact_5568_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F2,filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_nu(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F2),G),Xs),Xs))) ) ).

% sorted_map_same
tff(fact_5569_sorted__wrt__upt,axiom,
    ! [Mb: nat,Nb: nat] : sorted_wrt(nat,ord_less(nat),upt(Mb,Nb)) ).

% sorted_wrt_upt
tff(fact_5570_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_5571_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I5: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I5)) ) ) ).

% sorted_nths
tff(fact_5572_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_5573_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_nv(fun(list(A),A),fun(list(A),fun(A,$o)),G),Xs),Xs)) ) ).

% sorted_same
tff(fact_5574_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_5575_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_5576_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_5577_ranI,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),Mb,A3) = aa(A,option(A),some(A),B2) )
     => member(A,B2,ran(B,A,Mb)) ) ).

% ranI
tff(fact_5578_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)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_filter_le
tff(fact_5579_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)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),filter2(A,aTP_Lamp_nw(fun(A,$o),fun(A,$o),P),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_5580_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
        <=> sorted_wrt(B,aTP_Lamp_nx(fun(B,A),fun(B,fun(B,$o)),F2),Xs) ) ) ).

% sorted_map
tff(fact_5581_replicate__length__filter,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),filter2(A,aa(A,fun(A,$o),fequal(A),Xb),Xs)),Xb) = filter2(A,aa(A,fun(A,$o),fequal(A),Xb),Xs) ).

% replicate_length_filter
tff(fact_5582_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)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_filter_less
tff(fact_5583_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_5584_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_5585_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_5586_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),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),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5587_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_5588_map__of__filter__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,Z: A,P: fun(B,fun(A,$o))] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Z) )
     => ( aa(A,$o,aa(B,fun(A,$o),P,K),Z)
       => ( aa(B,option(A),map_of(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)),K) = aa(A,option(A),some(A),Z) ) ) ) ).

% map_of_filter_in
tff(fact_5589_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_5590_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,filter2(A,P,Xs)))),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,Xs))) ).

% sum_list_filter_le_nat
tff(fact_5591_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_ny(fun(B,option(A)),fun(A,$o),Mb)) ).

% ran_def
tff(fact_5592_sorted__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),map(product_prod(nat,A),nat,product_fst(nat,A),enumerate(A,Nb,Xs))) ).

% sorted_enumerate
tff(fact_5593_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_5594_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
             => ( aa(nat,$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),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5595_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( finite_finite(A,A4)
         => ? [X4: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X4) = A4 )
              & sorted_wrt(A,ord_less_eq(A),X4)
              & distinct(A,X4)
              & ! [Y5: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y5) = A4 )
                    & sorted_wrt(A,ord_less_eq(A),Y5)
                    & distinct(A,Y5) )
                 => ( Y5 = X4 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5596_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: 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_nz(fun(A,$o),fun(list(A),fun(nat,$o)),P),Xs))) ).

% filter_eq_nths
tff(fact_5597_length__filter__conv__card,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(list(A),nat,size_size(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_nz(fun(A,$o),fun(list(A),fun(nat,$o)),P2),Xs))) ).

% length_filter_conv_card
tff(fact_5598_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5599_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$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),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5600_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_5601_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( finite_finite(A,A4)
         => ~ ! [L3: list(A)] :
                ( sorted_wrt(A,ord_less(A),L3)
               => ( ( aa(list(A),set(A),set2(A),L3) = A4 )
                 => ( aa(list(A),nat,size_size(list(A)),L3) != aa(set(A),nat,finite_card(A),A4) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5602_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X5: A] :
                ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
                & aa(A,$o,P,X5) )
           => ( find(A,P,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_oa(list(A),fun(fun(A,$o),fun(A,$o)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_5603_nths__shift__lemma,axiom,
    ! [A: $tType,A4: set(nat),Xs: list(A),I: nat] : map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_ns(set(nat),fun(product_prod(A,nat),$o),A4),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)))))) = map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_ob(set(nat),fun(nat,fun(product_prod(A,nat),$o)),A4),I),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_5604_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,$o),Xs: list(B)] : map(B,A,F2,filter2(B,P,Xs)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_oc(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F2),P),Xs) ).

% map_filter_map_filter
tff(fact_5605_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S3)
       => ( finite_finite(B,A4)
         => ~ ! [L3: list(B)] :
                ( sorted_wrt(A,Less,map(B,A,F2,L3))
               => ( ( aa(list(B),set(B),set2(B),L3) = A4 )
                 => ( aa(list(B),nat,size_size(list(B)),L3) != aa(set(B),nat,finite_card(B),A4) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_5606_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),L: list(A)] :
          ( finite_finite(A,A4)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A4 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A4) ) )
          <=> ( linord4507533701916653071of_set(A,A4) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5607_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : aa(list(A),nat,size_size(list(A)),linord4507533701916653071of_set(A,A4)) = aa(set(A),nat,finite_card(A),A4) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_5608_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : sorted_wrt(A,ord_less_eq(A),linord4507533701916653071of_set(A,A4)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5609_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : sorted_wrt(A,ord_less(A),linord4507533701916653071of_set(A,A4)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5610_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)
           => ( linord4507533701916653071of_set(A,aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5611_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,$o)),S3: set(B),F2: fun(B,A),A4: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S3)
       => ( finite_finite(B,A4)
         => ( ( sorted_wrt(A,Less,map(B,A,F2,L))
              & ( aa(list(B),set(B),set2(B),L) = A4 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A4) ) )
          <=> ( sorted8670434370408473282of_set(A,B,Less_eq,F2,A4) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_5612_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = map(list(A),A,aTP_Lamp_od(nat,fun(list(A),A),I),filter2(list(A),aTP_Lamp_oe(nat,fun(list(A),$o),I),Xs)) ) ) ).

% nth_transpose
tff(fact_5613_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)),map(A,product_prod(A,A),aTP_Lamp_nh(A,product_prod(A,A)),Xs)) ).

% Id_on_set
tff(fact_5614_Id__onI,axiom,
    ! [A: $tType,A3: A,A4: set(A)] :
      ( member(A,A3,A4)
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),id_on(A,A4)) ) ).

% Id_onI
tff(fact_5615_Id__onE,axiom,
    ! [A: $tType,C2: product_prod(A,A),A4: set(A)] :
      ( member(product_prod(A,A),C2,id_on(A,A4))
     => ~ ! [X4: A] :
            ( member(A,X4,A4)
           => ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ) ) ).

% Id_onE
tff(fact_5616_Id__on__eqI,axiom,
    ! [A: $tType,A3: A,B2: A,A4: set(A)] :
      ( ( A3 = B2 )
     => ( member(A,A3,A4)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),id_on(A,A4)) ) ) ).

% Id_on_eqI
tff(fact_5617_Id__on__iff,axiom,
    ! [A: $tType,Xb: A,Y: A,A4: 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,A4))
    <=> ( ( Xb = Y )
        & member(A,Xb,A4) ) ) ).

% Id_on_iff
tff(fact_5618_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = foldr(list(A),nat,aTP_Lamp_of(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_5619_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,$o)),S3: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S3)
       => ( aa(list(B),nat,size_size(list(B)),sorted8670434370408473282of_set(A,B,Less_eq,F2,A4)) = aa(set(B),nat,finite_card(B),A4) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_5620_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,$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,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_oe(nat,fun(list(A),$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_5621_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( map(list(A),A,aTP_Lamp_od(nat,fun(list(A),A),I),filter2(list(A),aTP_Lamp_oe(nat,fun(list(A),$o),I),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_5622_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) = map(nat,list(A),aTP_Lamp_oh(list(list(A)),fun(nat,list(A)),Xs),upt(zero_zero(nat),Nb)) ) ) ) ).

% transpose_rectangle
tff(fact_5623_length__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),rev(A,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rev
tff(fact_5624_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_5625_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_5626_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,rev(A,Xs),rev(B,Ys)) = rev(product_prod(A,B),zip(A,B,Xs,Ys)) ) ) ).

% zip_rev
tff(fact_5627_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_5628_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_5629_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_5630_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,$o)] : find(A,Uu,nil(A)) = none(A) ).

% find.simps(1)
tff(fact_5631_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( 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_5632_rotate__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Nb),rev(A,Xs)) = 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_5633_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F2,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs))) ) ) ) ).

% sum_list_strict_mono
tff(fact_5634_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(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_5635_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,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_5636_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( rev(A,list_update(A,Xs,K,Y)) = list_update(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)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_5637_sorted__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_5638_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))),aa(nat,A,nth(A,Xs),I4)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_5639_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( aa(nat,$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_5640_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$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),I4)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_5641_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),foldr(list(B),nat,aTP_Lamp_oi(list(B),fun(nat,nat)),Xss,zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),foldr(list(B),nat,aTP_Lamp_oj(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_ok(list(B),$o),Xss),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_5642_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_of(list(A),fun(nat,nat)),transpose(A,Xs),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_ol(list(A),$o),Xs)) ).

% transpose_max_length
tff(fact_5643_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_oe(nat,fun(list(A),$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_5644_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_ol(list(A),$o),Xs) ) ) ).

% transpose_transpose
tff(fact_5645_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Ta: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,map(B,A,F2,Xs)))
         => ( filter2(B,aa(A,fun(B,$o),aTP_Lamp_om(fun(B,A),fun(A,fun(B,$o)),F2),Ta),Xs) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_om(fun(B,A),fun(A,fun(B,$o)),F2),Ta),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_5646_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P2: fun(A,B),I: A] :
          ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_as(set(A),fun(fun(A,B),fun(A,$o)),I5),P2)))
         => ( groups1027152243600224163dd_sum(A,B,P2,aa(set(A),set(A),insert(A,I),I5)) = $ite(member(A,I,I5),groups1027152243600224163dd_sum(A,B,P2,I5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P2,I)),groups1027152243600224163dd_sum(A,B,P2,I5))) ) ) ) ).

% sum.insert'
tff(fact_5647_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_5648_length__concat__rev,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,rev(list(A),Xs))) = aa(list(A),nat,size_size(list(A)),concat(A,Xs)) ).

% length_concat_rev
tff(fact_5649_map__of__eq__empty__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ! [X: A] : aa(A,option(B),map_of(A,B,Xys),X) = none(B)
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% map_of_eq_empty_iff
tff(fact_5650_empty__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ( aTP_Lamp_on(A,option(B)) = map_of(A,B,Xys) )
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% empty_eq_map_of_iff
tff(fact_5651_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_5652_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_5653_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_5654_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_5655_map__of_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType,X5: A] : aa(A,option(B),map_of(A,B,nil(product_prod(A,B))),X5) = none(B) ).

% map_of.simps(1)
tff(fact_5656_map__of__Cons__code_I1_J,axiom,
    ! [B: $tType,A: $tType,K: B] : aa(B,option(A),map_of(B,A,nil(product_prod(B,A))),K) = none(A) ).

% map_of_Cons_code(1)
tff(fact_5657_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite(A,I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oo(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_5658_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_5659_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_5660_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_as(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_as(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oo(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_5661_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_5662_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_5663_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_op(nat,fun(nat,nat),Nb),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ) ).

% range_mod
tff(fact_5664_cauchyD,axiom,
    ! [X6: fun(nat,rat),R: rat] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R)
       => ? [K3: nat] :
          ! [M: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),M)
           => ! [N5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N5)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M)),aa(nat,rat,X6,N5)))),R) ) ) ) ) ).

% cauchyD
tff(fact_5665_cauchyI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R3)
         => ? [K4: nat] :
            ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),M2)
             => ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),N)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M2)),aa(nat,rat,X6,N)))),R3) ) ) )
     => aa(fun(nat,rat),$o,cauchy,X6) ) ).

% cauchyI
tff(fact_5666_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(option(A),top_top(set(option(A))))
    <=> finite_finite(A,top_top(set(A))) ) ).

% finite_option_UNIV
tff(fact_5667_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),A3),top_top(set(A))) = top_top(set(A)) ) ).

% range_add
tff(fact_5668_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),A3),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_5669_surj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( ( image(A,A,F2,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),F2),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_5670_cauchy__diff,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(fun(nat,rat),$o,cauchy,Y6)
       => aa(fun(nat,rat),$o,cauchy,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ).

% cauchy_diff
tff(fact_5671_cauchy__minus,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => aa(fun(nat,rat),$o,cauchy,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_or(fun(nat,rat),fun(nat,rat)),X6)) ) ).

% cauchy_minus
tff(fact_5672_cauchy__mult,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(fun(nat,rat),$o,cauchy,Y6)
       => aa(fun(nat,rat),$o,cauchy,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_os(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% cauchy_mult
tff(fact_5673_cauchy__const,axiom,
    ! [Xb: rat] : aa(fun(nat,rat),$o,cauchy,aTP_Lamp_ot(rat,fun(nat,rat),Xb)) ).

% cauchy_const
tff(fact_5674_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A3) ) ).

% top.extremum_strict
tff(fact_5675_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( ( A3 != top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_5676_cauchy__add,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(fun(nat,rat),$o,cauchy,Y6)
       => aa(fun(nat,rat),$o,cauchy,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ou(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% cauchy_add
tff(fact_5677_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A3)
         => ( A3 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_5678_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A3)
        <=> ( A3 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_5679_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),top_top(A)) ) ).

% top_greatest
tff(fact_5680_bij__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_5681_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_5682_finite__range__Some,axiom,
    ! [A: $tType] :
      ( finite_finite(option(A),image(A,option(A),some(A),top_top(set(A))))
    <=> finite_finite(A,top_top(set(A))) ) ).

% finite_range_Some
tff(fact_5683_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_5684_cauchy__imp__bounded,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ? [B3: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B3)
          & ! [N5: nat] : aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N5))),B3) ) ) ).

% cauchy_imp_bounded
tff(fact_5685_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_ov(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_5686_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( finite_finite(A,top_top(set(A)))
     => aa(nat,$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_5687_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( finite_finite(A,image(B,A,F2,top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),image(B,A,F2,top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_5688_cauchy__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
    <=> ! [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
         => ? [K2: nat] :
            ! [M5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),M5)
             => ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N4)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M5)),aa(nat,rat,X6,N4)))),R5) ) ) ) ) ).

% cauchy_def
tff(fact_5689_le__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(fun(nat,rat),$o,cauchy,Y6)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y6))
        <=> ! [R5: rat] :
              ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
             => ? [K2: nat] :
                ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N4)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X6,N4)),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Y6,N4)),R5)) ) ) ) ) ) ).

% le_Real
tff(fact_5690_cauchy__not__vanishes,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B3)
            & ? [K3: nat] :
              ! [N5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N5)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B3),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N5))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_5691_sums__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] : sums(A,F2,aa(set(A),A,complete_Sup_Sup(A),image(nat,A,aTP_Lamp_ow(fun(nat,A),fun(nat,A),F2),top_top(set(nat))))) ) ).

% sums_SUP
tff(fact_5692_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_5693_vanishes__const,axiom,
    ! [C2: rat] :
      ( vanishes(aTP_Lamp_ot(rat,fun(nat,rat),C2))
    <=> ( C2 = zero_zero(rat) ) ) ).

% vanishes_const
tff(fact_5694_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_5695_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_5696_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_5697_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_5698_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_5699_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_5700_eq__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(fun(nat,rat),$o,cauchy,Y6)
       => ( ( aa(fun(nat,rat),real,real2,X6) = aa(fun(nat,rat),real,real2,Y6) )
        <=> vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ) ).

% eq_Real
tff(fact_5701_inverse__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(real,real,inverse_inverse(real),aa(fun(nat,rat),real,real2,X6)) = $ite(vanishes(X6),zero_zero(real),aa(fun(nat,rat),real,real2,aTP_Lamp_ox(fun(nat,rat),fun(nat,rat),X6))) ) ) ).

% inverse_Real
tff(fact_5702_cauchy__inverse,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( ~ vanishes(X6)
       => aa(fun(nat,rat),$o,cauchy,aTP_Lamp_ox(fun(nat,rat),fun(nat,rat),X6)) ) ) ).

% cauchy_inverse
tff(fact_5703_Real__induct,axiom,
    ! [P: fun(real,$o),Xb: real] :
      ( ! [X8: fun(nat,rat)] :
          ( aa(fun(nat,rat),$o,cauchy,X8)
         => aa(real,$o,P,aa(fun(nat,rat),real,real2,X8)) )
     => aa(real,$o,P,Xb) ) ).

% Real_induct
tff(fact_5704_vanishes__minus,axiom,
    ! [X6: fun(nat,rat)] :
      ( vanishes(X6)
     => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_or(fun(nat,rat),fun(nat,rat)),X6)) ) ).

% vanishes_minus
tff(fact_5705_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A3) )
           => ( ! [Y4: A] :
                  ( ! [X5: A] :
                      ( member(A,X5,X6)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Y4) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Y4) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_5706_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_5707_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Xb: A] :
          ( finite_finite(A,X6)
         => ( member(A,Xb,X6)
           => aa(A,$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_5708_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_5709_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_5710_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Xb: A,A3: A] :
          ( finite_finite(A,X6)
         => ( member(A,Xb,X6)
           => ( ! [X4: A] :
                  ( member(A,X4,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A3) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A3) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_5711_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_5712_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X6: set(A),A3: A] :
          ( ! [X4: A] :
              ( member(A,X4,X6)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A3) )
         => ( ! [Y4: A] :
                ( ! [X5: A] :
                    ( member(A,X5,X6)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Y4) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Y4) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ).

% cSup_eq
tff(fact_5713_of__nat__Real,axiom,
    ! [Xb: nat] : aa(nat,real,semiring_1_of_nat(real),Xb) = aa(fun(nat,rat),real,real2,aTP_Lamp_oy(nat,fun(nat,rat),Xb)) ).

% of_nat_Real
tff(fact_5714_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S3: set(C),X5: A,Xa: B] :
      ( aa(B,$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,$o)),aTP_Lamp_oz(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R),S3)),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),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)),R,S3))) ) ).

% SUP_UN_eq2
tff(fact_5715_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X5: A,Xa: B] :
      ( aa(B,$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,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),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_5716_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(fun(A,fun(B,$o))),X5: 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,$o))),S3),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image(fun(product_prod(A,B),$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_5717_vanishes__add,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( vanishes(X6)
     => ( vanishes(Y6)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ou(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% vanishes_add
tff(fact_5718_card__Union__le__sum__card,axiom,
    ! [A: $tType,U2: set(set(A))] : aa(nat,$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)),U2))),aa(set(set(A)),nat,groups7311177749621191930dd_sum(set(A),nat,finite_card(A)),U2)) ).

% card_Union_le_sum_card
tff(fact_5719_vanishes__diff,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( vanishes(X6)
     => ( vanishes(Y6)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ).

% vanishes_diff
tff(fact_5720_zero__real__def,axiom,
    zero_zero(real) = aa(fun(nat,rat),real,real2,aTP_Lamp_pa(nat,rat)) ).

% zero_real_def
tff(fact_5721_mult__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(fun(nat,rat),$o,cauchy,Y6)
       => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_os(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ) ).

% mult_Real
tff(fact_5722_one__real__def,axiom,
    one_one(real) = aa(fun(nat,rat),real,real2,aTP_Lamp_pb(nat,rat)) ).

% one_real_def
tff(fact_5723_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),M6: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),M6) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))),M6) ) ) ) ).

% cSUP_least
tff(fact_5724_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A3: A] :
          ( finite_finite(A,X6)
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A3)
            <=> ! [X: A] :
                  ( member(A,X,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_5725_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X4)),A3) )
           => aa(A,$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))),A3) ) ) ) ).

% cSup_abs_le
tff(fact_5726_of__int__Real,axiom,
    ! [Xb: int] : aa(int,real,ring_1_of_int(real),Xb) = aa(fun(nat,rat),real,real2,aTP_Lamp_pc(int,fun(nat,rat),Xb)) ).

% of_int_Real
tff(fact_5727_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U2: set(set(A))] :
      ( ! [X4: set(A)] :
          ( member(set(A),X4,U2)
         => finite_finite(A,X4) )
     => aa(nat,$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)),U2))),aa(set(set(A)),nat,groups7311177749621191930dd_sum(set(A),nat,finite_card(A)),U2)) ) ).

% card_Union_le_sum_card_weak
tff(fact_5728_vanishes__diff__inverse,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( ~ vanishes(X6)
       => ( aa(fun(nat,rat),$o,cauchy,Y6)
         => ( ~ vanishes(Y6)
           => ( vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6))
             => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_pd(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ) ) ) ).

% vanishes_diff_inverse
tff(fact_5729_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_pe(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K),set_ord_lessThan(nat,Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M6,set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ).

% UN_le_add_shift_strict
tff(fact_5730_UN__le__add__shift,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_pe(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K),set_ord_atMost(nat,Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M6,set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ).

% UN_le_add_shift
tff(fact_5731_Id__on__def,axiom,
    ! [A: $tType,A4: set(A)] : id_on(A,A4) = 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_pf(A,set(product_prod(A,A))),A4)) ).

% Id_on_def
tff(fact_5732_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),L))),E) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S3)),L))),E) ) ) ) ).

% cSup_asclose
tff(fact_5733_minus__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(real,real,uminus_uminus(real),aa(fun(nat,rat),real,real2,X6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_or(fun(nat,rat),fun(nat,rat)),X6)) ) ) ).

% minus_Real
tff(fact_5734_add__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(fun(nat,rat),$o,cauchy,Y6)
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ou(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ) ).

% add_Real
tff(fact_5735_diff__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(fun(nat,rat),$o,cauchy,Y6)
       => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ) ).

% diff_Real
tff(fact_5736_UN__finite2__eq,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A4,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B4,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A4,top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B4,top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_5737_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A4: fun(A,set(B))] :
      ( finite_finite(A,I5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),image(A,set(B),A4,I5)))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_pg(fun(A,set(B)),fun(A,nat),A4)),I5)) ) ).

% card_UN_le
tff(fact_5738_vanishes__mult__bounded,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( ? [A9: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A9)
          & ! [N: nat] : aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N))),A9) )
     => ( vanishes(Y6)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_os(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% vanishes_mult_bounded
tff(fact_5739_vanishes__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( vanishes(X6)
    <=> ! [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
         => ? [K2: nat] :
            ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N4)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N4))),R5) ) ) ) ).

% vanishes_def
tff(fact_5740_vanishesI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R3)
         => ? [K4: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),N)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N))),R3) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_5741_vanishesD,axiom,
    ! [X6: fun(nat,rat),R: rat] :
      ( vanishes(X6)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R)
       => ? [K3: nat] :
          ! [N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N5)
           => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N5))),R) ) ) ) ).

% vanishesD
tff(fact_5742_UN__finite2__subset,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A4,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B4,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A4,top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B4,top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_5743_suminf__eq__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = aa(set(A),A,complete_Sup_Sup(A),image(nat,A,aTP_Lamp_ow(fun(nat,A),fun(nat,A),F2),top_top(set(nat)))) ) ).

% suminf_eq_SUP
tff(fact_5744_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B3)
            & ? [K3: nat] :
                ( ! [N5: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N5)
                   => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B3),aa(rat,rat,uminus_uminus(rat),aa(nat,rat,X6,N5))) )
                | ! [N5: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N5)
                   => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B3),aa(nat,rat,X6,N5)) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_5745_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4)) = top_top(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),top_top(A))
             => ? [Xa2: B] :
                  ( member(B,Xa2,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(B,A,F2,Xa2)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_5746_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F2: fun(nat,set(A)),S3: set(A)] :
      ( ! [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F2,I2)),S3)
     => ( finite_finite(A,S3)
       => ( ? [N7: nat] :
              ( ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N7)
                 => ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N7)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F2,M2)),aa(nat,set(A),F2,N)) ) ) )
              & ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
                 => ( aa(nat,set(A),F2,N7) = aa(nat,set(A),F2,N) ) ) )
         => ( aa(nat,set(A),F2,aa(set(A),nat,finite_card(A),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),F2,top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_5747_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A4) = top_top(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),top_top(A))
             => ? [Xa2: A] :
                  ( member(A,Xa2,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa2) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_5748_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),Xb: A] :
          ( ! [Y4: A] :
              ( member(A,Y4,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
         => ( ! [Y4: A] :
                ( ! [Z4: A] :
                    ( member(A,Z4,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z4),Y4) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y4) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A4) = Xb ) ) ) ) ).

% Sup_eqI
tff(fact_5749_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B4: set(A)] :
          ( ! [A5: A] :
              ( member(A,A5,A4)
             => ? [X5: A] :
                  ( member(A,X5,B4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),X5) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B4)) ) ) ).

% Sup_mono
tff(fact_5750_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),Z: A] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(A,$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),A4)),Z) ) ) ).

% Sup_least
tff(fact_5751_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A4: set(A)] :
          ( member(A,Xb,A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ).

% Sup_upper
tff(fact_5752_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),B2)
        <=> ! [X: A] :
              ( member(A,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2) ) ) ) ).

% Sup_le_iff
tff(fact_5753_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A4: set(A),V: A] :
          ( member(A,U,A4)
         => ( aa(A,$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),A4)) ) ) ) ).

% Sup_upper2
tff(fact_5754_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,S3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),S3))
        <=> ? [X: A] :
              ( member(A,X,S3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X) ) ) ) ).

% less_Sup_iff
tff(fact_5755_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,A4: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A4))
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb)
             => ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X) ) ) ) ) ).

% le_Sup_iff
tff(fact_5756_SUP__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(A),B4: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => ? [X5: B] :
                  ( member(B,X5,B4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,I2)),aa(B,C,G,X5)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B4)
               => ? [X5: A] :
                    ( member(A,X5,A4)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,J2)),aa(A,C,F2,X5)) ) )
           => ( aa(set(C),C,complete_Sup_Sup(C),image(A,C,F2,A4)) = aa(set(C),C,complete_Sup_Sup(C),image(B,C,G,B4)) ) ) ) ) ).

% SUP_eq
tff(fact_5757_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V3) )
         => ( ( A4 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ).

% less_eq_Sup
tff(fact_5758_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B4)) ) ) ).

% Sup_subset_mono
tff(fact_5759_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),U: B,F2: fun(A,B)] :
          ( member(A,I,A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))) ) ) ) ).

% SUP_upper2
tff(fact_5760_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A4: set(B),U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))),U)
        <=> ! [X: B] :
              ( member(B,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X)),U) ) ) ) ).

% SUP_le_iff
tff(fact_5761_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),F2: fun(A,B)] :
          ( member(A,I,A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))) ) ) ).

% SUP_upper
tff(fact_5762_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A4: set(A)] :
          ( ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))),aa(set(B),B,complete_Sup_Sup(B),image(A,B,G,A4))) ) ) ).

% SUP_mono'
tff(fact_5763_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: fun(A,B),U: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),U) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))),U) ) ) ).

% SUP_least
tff(fact_5764_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(A),B4: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [N: A] :
              ( member(A,N,A4)
             => ? [X5: B] :
                  ( member(B,X5,B4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,N)),aa(B,C,G,X5)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),image(A,C,F2,A4))),aa(set(C),C,complete_Sup_Sup(C),image(B,C,G,B4))) ) ) ).

% SUP_mono
tff(fact_5765_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: fun(A,B),Xb: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),Xb) )
         => ( ! [Y4: B] :
                ( ! [I3: A] :
                    ( member(A,I3,A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),Y4) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),Y4) )
           => ( aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4)) = Xb ) ) ) ) ).

% SUP_eqI
tff(fact_5766_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,F2: fun(B,A),A4: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4)))
        <=> ? [X: B] :
              ( member(B,X,A4)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,X)) ) ) ) ).

% less_SUP_iff
tff(fact_5767_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A4: set(B),Y: A,I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))),Y)
         => ( member(B,I,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,I)),Y) ) ) ) ).

% SUP_lessD
tff(fact_5768_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,F2: fun(B,A),A4: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4)))
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb)
             => ? [X: B] :
                  ( member(B,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),aa(B,A,F2,X)) ) ) ) ) ).

% le_SUP_iff
tff(fact_5769_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),C2: B,F2: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(A,B,F2,I2)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,I5)) = C2 )
            <=> ! [X: A] :
                  ( member(A,X,I5)
                 => ( aa(A,B,F2,X) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_5770_SUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),B4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
         => ( ! [X4: A] :
                ( member(A,X4,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))),aa(set(B),B,complete_Sup_Sup(B),image(A,B,G,B4))) ) ) ) ).

% SUP_subset_mono
tff(fact_5771_card__UNION,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( finite_finite(set(A),A4)
     => ( ! [X4: set(A)] :
            ( member(set(A),X4,A4)
           => finite_finite(A,X4) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)) = aa(int,nat,nat2,aa(set(set(set(A))),int,groups7311177749621191930dd_sum(set(set(A)),int,aTP_Lamp_ph(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_pi(set(set(A)),fun(set(set(A)),$o),A4)))) ) ) ) ).

% card_UNION
tff(fact_5772_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_pj(nat,fun(real,real),Nb),Xb)) ).

% root_def
tff(fact_5773_not__positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( ~ aa(real,$o,positive2,aa(fun(nat,rat),real,real2,X6))
      <=> ! [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
           => ? [K2: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N4)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X6,N4)),R5) ) ) ) ) ).

% not_positive_Real
tff(fact_5774_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A4) = bot_bot(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X)
             => ? [Xa2: A] :
                  ( member(A,Xa2,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_5775_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_5776_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_5777_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_5778_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_5779_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4)) = bot_bot(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X)
             => ? [Xa2: B] :
                  ( member(B,Xa2,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Xa2)),X) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_5780_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_5781_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V3),U) )
         => ( ( A4 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),U) ) ) ) ).

% Inf_less_eq
tff(fact_5782_INF__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(A),B4: set(B),G: fun(B,C),F2: fun(A,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => ? [X5: B] :
                  ( member(B,X5,B4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,X5)),aa(A,C,F2,I2)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B4)
               => ? [X5: A] :
                    ( member(A,X5,A4)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X5)),aa(B,C,G,J2)) ) )
           => ( aa(set(C),C,complete_Inf_Inf(C),image(A,C,F2,A4)) = aa(set(C),C,complete_Inf_Inf(C),image(B,C,G,B4)) ) ) ) ) ).

% INF_eq
tff(fact_5783_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S3: set(A),A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S3)),A3)
        <=> ? [X: A] :
              ( member(A,X,S3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3) ) ) ) ).

% Inf_less_iff
tff(fact_5784_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),Xb)
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y3)
             => ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) ) ) ) ) ).

% Inf_le_iff
tff(fact_5785_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),Xb: A] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),I2) )
         => ( ! [Y4: A] :
                ( ! [I3: A] :
                    ( member(A,I3,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),I3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A4) = Xb ) ) ) ) ).

% Inf_eqI
tff(fact_5786_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A4: set(A)] :
          ( ! [B3: A] :
              ( member(A,B3,B4)
             => ? [X5: A] :
                  ( member(A,X5,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),B3) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B4)) ) ) ).

% Inf_mono
tff(fact_5787_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A4: set(A)] :
          ( member(A,Xb,A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),Xb) ) ) ).

% Inf_lower
tff(fact_5788_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A4: set(A),V: A] :
          ( member(A,U,A4)
         => ( aa(A,$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),A4)),V) ) ) ) ).

% Inf_lower2
tff(fact_5789_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A4: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A4))
        <=> ! [X: A] :
              ( member(A,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X) ) ) ) ).

% le_Inf_iff
tff(fact_5790_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),Z: A] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(A,$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),A4)) ) ) ).

% Inf_greatest
tff(fact_5791_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B4)) ) ) ).

% Inf_superset_mono
tff(fact_5792_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X4) )
           => ( ! [Y4: A] :
                  ( ! [X5: A] :
                      ( member(A,X5,X6)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X5) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),A3) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_5793_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_5794_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_5795_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X6: set(A),A3: A] :
          ( ! [X4: A] :
              ( member(A,X4,X6)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X4) )
         => ( ! [Y4: A] :
                ( ! [X5: A] :
                    ( member(A,X5,X6)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X5) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),A3) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ).

% cInf_eq
tff(fact_5796_Real_Opositive__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,positive2,Xb)
     => ( aa(real,$o,positive2,Y)
       => aa(real,$o,positive2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% Real.positive_add
tff(fact_5797_Real_Opositive__zero,axiom,
    ~ aa(real,$o,positive2,zero_zero(real)) ).

% Real.positive_zero
tff(fact_5798_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_5799_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Xb: A,A3: A] :
          ( finite_finite(A,X6)
         => ( member(A,Xb,X6)
           => ( ! [X4: A] :
                  ( member(A,X4,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_5800_Real_Opositive__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,positive2,Xb)
     => ( aa(real,$o,positive2,Y)
       => aa(real,$o,positive2,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) ) ) ).

% Real.positive_mult
tff(fact_5801_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Xb: A] :
          ( finite_finite(A,X6)
         => ( member(A,Xb,X6)
           => aa(A,$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_5802_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),Xb: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),aa(A,B,F2,I2)) )
         => ( ! [Y4: B] :
                ( ! [I3: A] :
                    ( member(A,I3,A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y4),aa(A,B,F2,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y4),Xb) )
           => ( aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4)) = Xb ) ) ) ) ).

% INF_eqI
tff(fact_5803_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B4: set(A),A4: set(B),F2: fun(B,C),G: fun(A,C)] :
          ( ! [M2: A] :
              ( member(A,M2,B4)
             => ? [X5: B] :
                  ( member(B,X5,A4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F2,X5)),aa(A,C,G,M2)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),image(B,C,F2,A4))),aa(set(C),C,complete_Inf_Inf(C),image(A,C,G,B4))) ) ) ).

% INF_mono
tff(fact_5804_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),F2: fun(A,B)] :
          ( member(A,I,A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))),aa(A,B,F2,I)) ) ) ).

% INF_lower
tff(fact_5805_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A4: set(A)] :
          ( ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,G,A4))) ) ) ).

% INF_mono'
tff(fact_5806_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),F2: fun(A,B),U: B] :
          ( member(A,I,A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))),U) ) ) ) ).

% INF_lower2
tff(fact_5807_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F2: fun(B,A),A4: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4)))
        <=> ! [X: B] :
              ( member(B,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F2,X)) ) ) ) ).

% le_INF_iff
tff(fact_5808_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),U: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))) ) ) ).

% INF_greatest
tff(fact_5809_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F2: fun(B,A),A4: set(B),I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4)))
         => ( member(B,I,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,I)) ) ) ) ).

% less_INF_D
tff(fact_5810_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B),A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))),A3)
        <=> ? [X: B] :
              ( member(B,X,A4)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),A3) ) ) ) ).

% INF_less_iff
tff(fact_5811_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))),Xb)
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y3)
             => ? [X: B] :
                  ( member(B,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),Y3) ) ) ) ) ).

% INF_le_iff
tff(fact_5812_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),Mb: B,F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(A,B,F2,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))) ) ) ) ).

% cINF_greatest
tff(fact_5813_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),C2: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),C2) )
           => ( ( aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,I5)) = C2 )
            <=> ! [X: A] :
                  ( member(A,X,I5)
                 => ( aa(A,B,F2,X) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_5814_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A3: A] :
          ( finite_finite(A,X6)
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6))
            <=> ! [X: A] :
                  ( member(A,X,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_5815_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ).

% Inf_le_Sup
tff(fact_5816_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X4)),A3) )
           => aa(A,$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))),A3) ) ) ) ).

% cInf_abs_ge
tff(fact_5817_INF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B4: set(A),A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A4)
         => ( ! [X4: A] :
                ( member(A,X4,B4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,G,B4))) ) ) ) ).

% INF_superset_mono
tff(fact_5818_Real_Opositive__minus,axiom,
    ! [Xb: real] :
      ( ~ aa(real,$o,positive2,Xb)
     => ( ( Xb != zero_zero(real) )
       => aa(real,$o,positive2,aa(real,real,uminus_uminus(real),Xb)) ) ) ).

% Real.positive_minus
tff(fact_5819_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A4) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_pk(set(A),fun(A,$o),A4))) ) ).

% Inf_eq_Sup
tff(fact_5820_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A4) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_pl(set(A),fun(A,$o),A4))) ) ).

% Sup_eq_Inf
tff(fact_5821_less__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
    <=> aa(real,$o,positive2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),Xb)) ) ).

% less_real_def
tff(fact_5822_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))) ) ) ).

% INF_le_SUP
tff(fact_5823_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),L))),E) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S3)),L))),E) ) ) ) ).

% cInf_asclose
tff(fact_5824_positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(real,$o,positive2,aa(fun(nat,rat),real,real2,X6))
      <=> ? [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
            & ? [K2: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N4)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,X6,N4)) ) ) ) ) ).

% positive_Real
tff(fact_5825_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_5826_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_5827_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D5: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( Xb != zero_zero(real) )
       => ( ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
             => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
               => ( D5 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
               => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Nb),D5,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_5828_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_5829_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))),aa(real,real,F2,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_5830_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_5831_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4))),aa(real,real,F2,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_5832_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Xb: A,Z: A] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_pm(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_5833_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Mb: A,Xb: A] :
          ( has_field_derivative(A,G,Mb,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_pn(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,aa(A,A,G,Xb))),Mb),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_5834_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S: set(A),G: fun(A,A),G5: fun(A,A),F2: fun(A,A),F6: A,Xb: A] :
          ( ! [X4: A] :
              ( member(A,X4,S)
             => has_field_derivative(A,G,aa(A,A,G5,X4),topolo174197925503356063within(A,X4,top_top(set(A)))) )
         => ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => ( member(A,aa(A,A,F2,Xb),S)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_po(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G5,aa(A,A,F2,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_5835_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G5: fun(A,A),F2: fun(A,A),F6: A,Xb: A] :
          ( ! [X4: A] : has_field_derivative(A,G,aa(A,A,G5,X4),topolo174197925503356063within(A,X4,top_top(set(A))))
         => ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_po(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G5,aa(A,A,F2,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_5836_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xb: A,Db: A,S: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_po(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_chain2
tff(fact_5837_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,aa(A,A,F2,Xb),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pp(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),E5),D5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_chain'
tff(fact_5838_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Mb: A,Xb: A] :
          ( has_field_derivative(A,G,Mb,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_pq(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G,Xb))),Mb),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_5839_DERIV__pos__imp__increasing,axiom,
    ! [A3: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y5) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,A3)),aa(real,real,F2,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_5840_DERIV__neg__imp__decreasing,axiom,
    ! [A3: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A3)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_5841_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_5842_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,Xb)),E5))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,Xb)),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_5843_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,Xb: A,S: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C2),C2,topolo174197925503356063within(A,Xb,S)) ) ).

% DERIV_cmult_Id
tff(fact_5844_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_ps(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D5),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_cmult
tff(fact_5845_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_pt(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),D5),C2),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_cmult_right
tff(fact_5846_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aTP_Lamp_pu(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,Xb))),Db),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_field_derivative_sinh
tff(fact_5847_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aTP_Lamp_pv(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,Xb))),Db),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_field_derivative_cosh
tff(fact_5848_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pw(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,Xb)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult'
tff(fact_5849_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,Xb: A,S: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pw(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult
tff(fact_5850_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( 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),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_5851_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( 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),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))),aa(real,real,F2,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_5852_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_px(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D5),C2),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_cdivide
tff(fact_5853_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4),S3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_5854_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4),S3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4))),aa(real,real,F2,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_5855_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F4: filter(A)] : has_field_derivative(A,aTP_Lamp_py(A,A),one_one(A),F4) ) ).

% DERIV_ident
tff(fact_5856_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pz(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D5),E5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_add
tff(fact_5857_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,F4: filter(A),G: fun(A,A),G5: A] :
          ( has_field_derivative(A,F2,F6,F4)
         => ( has_field_derivative(A,G,G5,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pz(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F6),G5),F4) ) ) ) ).

% field_differentiable_add
tff(fact_5858_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,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_qa(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_at_within_shift
tff(fact_5859_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_qb(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xb))),D5)),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xb)))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_5860_MVT2,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),F6: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
             => has_field_derivative(real,F2,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),A3),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,F2,B2)),aa(real,real,F2,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)),aa(real,real,F6,Z3)) ) ) ) ) ).

% MVT2
tff(fact_5861_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y4))),D2)
             => ( aa(real,real,F2,Xb) = aa(real,real,F2,Y4) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_5862_DERIV__ln,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_5863_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Mb: A,Xb: A] :
          ( has_field_derivative(A,G,Mb,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_qc(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,G,Xb)))),Mb),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_5864_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xaa: A] : has_field_derivative(A,aTP_Lamp_qd(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xaa),K))),topolo174197925503356063within(A,Xaa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_5865_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_qe(fun(A,A),fun(nat,fun(A,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Nb))),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xb)),Nb))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_power_Suc
tff(fact_5866_DERIV__const__average,axiom,
    ! [A3: real,B2: real,V: fun(real,real),K: real] :
      ( ( A3 != B2 )
     => ( ! [X4: real] : has_field_derivative(real,V,K,topolo174197925503356063within(real,X4,top_top(set(real))))
       => ( aa(real,real,V,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),B2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,V,A3)),aa(real,real,V,B2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ) ).

% DERIV_const_average
tff(fact_5867_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,S: 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,S)) ) ) ).

% DERIV_inverse
tff(fact_5868_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_qf(fun(A,A),fun(nat,fun(A,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_power
tff(fact_5869_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y4))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Xb)),aa(real,real,F2,Y4)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_5870_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y4))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Y4)),aa(real,real,F2,Xb)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_5871_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_5872_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_qg(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,S)) ).

% DERIV_pow
tff(fact_5873_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [Y4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C2),Y4))
         => has_field_derivative(A,aTP_Lamp_qh(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_5874_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo174197925503356063within(A,Xb,set_or1337092689740270186AtMost(A,A3,B2)) = topolo174197925503356063within(A,Xb,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_5875_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),Mb: real,Xb: real,Nb: nat] :
      ( has_field_derivative(real,G,Mb,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_qi(fun(real,real),fun(nat,fun(real,real)),G),Nb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,G,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))))),Mb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_5876_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( topolo174197925503356063within(A,B2,set_or1337092689740270186AtMost(A,A3,B2)) = topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_5877_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xb: A,S: set(A),G: fun(A,A),E: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),E),aa(A,A,F2,Xb)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_5878_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_qb(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_5879_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X5: A,Xa: B] :
      ( aa(B,$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,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),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_5880_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),F2: fun(A,A),F6: A,Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K5)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C2),Z3),aa(A,A,F2,Z3)) )
         => ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),C2),Z),F6) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_5881_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_qj(real,fun(real,real),R),aa(real,real,aa(real,fun(real,real),times_times(real),R),powr(real,Z,aa(real,real,aa(real,fun(real,real),minus_minus(real),R),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_5882_Union__maximal__sets,axiom,
    ! [A: $tType,F7: set(set(A))] :
      ( finite_finite(set(A),F7)
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_qk(set(set(A)),fun(set(A),$o),F7))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F7) ) ) ).

% Union_maximal_sets
tff(fact_5883_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S3: set(C),X5: A,Xa: B] :
      ( aa(B,$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,$o)),aTP_Lamp_oz(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R),S3)),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),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)),R,S3))) ) ).

% INF_INT_eq2
tff(fact_5884_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ql(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
               => has_field_derivative(A,aTP_Lamp_qh(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_5885_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => has_field_derivative(A,aTP_Lamp_qh(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_5886_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K5)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
           => has_field_derivative(A,aTP_Lamp_qh(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_5887_DERIV__log,axiom,
    ! [Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_5888_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),Mb: real,Xb: real,R: real] :
      ( has_field_derivative(real,G,Mb,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,Xb))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_qm(fun(real,real),fun(real,fun(real,real)),G),R),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),powr(real,aa(real,real,G,Xb),aa(real,real,aa(real,fun(real,real),minus_minus(real),R),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),Mb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_5889_DERIV__powr,axiom,
    ! [G: fun(real,real),Mb: real,Xb: real,F2: fun(real,real),R: real] :
      ( has_field_derivative(real,G,Mb,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,Xb))
       => ( has_field_derivative(real,F2,R,topolo174197925503356063within(real,Xb,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_qn(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,Xb),aa(real,real,F2,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),aa(real,real,ln_ln(real),aa(real,real,G,Xb)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),Mb),aa(real,real,F2,Xb))),aa(real,real,G,Xb)))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_5890_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_5891_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_5892_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_5893_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: set(real)] : has_field_derivative(real,arsinh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,Xb,A4)) ).

% arsinh_real_has_field_derivative
tff(fact_5894_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X: product_prod(A,A)] :
            ( member(product_prod(A,A),X,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys)))
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),fequal(A)),X) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_5895_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_5896_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A4: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),A4))),aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_qo(set(set(A)),fun(set(A),$o),A4))))) ) ).

% Inf_Sup_le
tff(fact_5897_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(set(A))] : aa(A,$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_qp(set(set(A)),fun(set(A),$o),A4))))),aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),A4))) ) ).

% Sup_Inf_le
tff(fact_5898_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Xb: A,Db: A,S: set(A)] :
          ( ( cosh(A,aa(A,A,G,Xb)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aTP_Lamp_qq(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),tanh(A,aa(A,A,G,Xb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Db),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_5899_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D5: real] :
      ( ( Xb != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( D5 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
       => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
           => ( D5 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
         => has_field_derivative(real,sqrt,D5,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_5900_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => has_field_derivative(real,arcosh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,Xb,A4)) ) ).

% arcosh_real_has_field_derivative
tff(fact_5901_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => has_field_derivative(real,artanh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(real,Xb,A4)) ) ).

% artanh_real_has_field_derivative
tff(fact_5902_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),$o),aTP_Lamp_qr(list(A),fun(list(A),$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_5903_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),$o),aTP_Lamp_qr(list(A),fun(list(A),$o))),X4) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_5904_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_5905_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_5906_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_5907_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(fun(A,fun(B,$o))),X5: 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,$o))),S3),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),image(fun(product_prod(A,B),$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_5908_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xb: real,Nb: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M2: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),X4),topolo174197925503356063within(real,X4,top_top(set(real))))
       => ? [T3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_qs(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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_5909_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xb: real,Nb: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M2: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),X4),topolo174197925503356063within(real,X4,top_top(set(real)))) )
     => ? [T3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),Xb))
          & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_qs(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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),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_5910_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_5911_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M2: nat,T3: 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)),T3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
           => ? [T3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),H)
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qt(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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_5912_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M2: nat,T3: 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)),T3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
         => ? [T3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T3)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),H)
              & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qt(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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_5913_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),zero_zero(real))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M2: nat,T3: 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),H),T3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
           => ? [T3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),T3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),zero_zero(real))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qt(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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_5914_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( Xb != zero_zero(real) )
         => ( ! [M2: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),X4),topolo174197925503356063within(real,X4,top_top(set(real))))
           => ? [T3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T3))
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),Xb))
                & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_qs(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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_5915_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M2: nat,T3: 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),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),Xb)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
       => ? [T3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_qs(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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_5916_Taylor,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B2: real,C2: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M2: nat,T3: 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),A3),T3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),B2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Xb)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),B2)
                 => ( ( Xb != C2 )
                   => ? [T3: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),C2),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T3)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),C2) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T3)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),Xb) ) )
                        & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qu(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),C2)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_5917_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B2: real,C2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M2: nat,T3: 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),A3),T3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),B2)
             => ? [T3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),B2)
                  & ( aa(real,real,F2,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qv(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_5918_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B2: real,C2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M2: nat,T3: 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),A3),T3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),B2)
             => ? [T3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),T3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T3),C2)
                  & ( aa(real,real,F2,A3) = 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_qv(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A3),C2)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Nb),T3)),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),A3),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_5919_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B4: real] :
      ( ! [M2: nat,T3: 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)),T3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T3),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
     => ( ( Nb = aa(nat,nat,suc,K) )
       => ! [M: nat,T6: 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)),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_qx(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Nb),Diff),B4),M),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),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_qy(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M),T6)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M))))),aa(real,real,aa(real,fun(real,real),times_times(real),B4),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),T6),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M)))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M))))))),topolo174197925503356063within(real,T6,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_5920_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_qz(real,real),suminf(real,aTP_Lamp_ra(real,fun(nat,real),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_5921_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A4: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),A4))),aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_rb(set(set(A)),fun(set(A),$o),A4))))) ) ).

% finite_Inf_Sup
tff(fact_5922_DERIV__power__series_H,axiom,
    ! [R2: real,F2: fun(nat,real),X0: real] :
      ( ! [X4: real] :
          ( member(real,X4,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_rc(fun(nat,real),fun(real,fun(nat,real)),F2),X4)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => has_field_derivative(real,aTP_Lamp_re(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_rc(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_5923_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_rf(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_rg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_5924_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_5925_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_5926_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_5927_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B2) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_5928_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ finite_finite(A,set_or5935395276787703475ssThan(A,A3,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% infinite_Ioo_iff
tff(fact_5929_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_5930_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_5931_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_5932_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_5933_has__field__derivative__imp__has__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,D5,F4)
         => has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D5),F4) ) ) ).

% has_field_derivative_imp_has_derivative
tff(fact_5934_has__derivative__imp__has__field__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: fun(A,A),F4: filter(A),D7: A] :
          ( has_derivative(A,A,F2,D5,F4)
         => ( ! [X4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X4),D7) = aa(A,A,D5,X4)
           => has_field_derivative(A,F2,D7,F4) ) ) ) ).

% has_derivative_imp_has_field_derivative
tff(fact_5935_has__field__derivative__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,D5,F4)
        <=> has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D5),F4) ) ) ).

% has_field_derivative_def
tff(fact_5936_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),F6: fun(A,real),Xb: A,S: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,real,F2,F6,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rh(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_ri(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),Xb),G),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_scaleR
tff(fact_5937_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F4: filter(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,F4)
         => ( has_derivative(A,B,G,G5,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_rj(fun(A,B),fun(fun(A,B),fun(A,B)),F6),G5),F4) ) ) ) ).

% has_derivative_add
tff(fact_5938_has__derivative__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G5: fun(A,B),F4: filter(A),Y: B] :
          ( has_derivative(A,B,G,G5,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_rk(fun(A,B),fun(B,fun(A,B)),G),Y),aa(B,fun(A,B),aTP_Lamp_rk(fun(A,B),fun(B,fun(A,B)),G5),Y),F4) ) ) ).

% has_derivative_mult_left
tff(fact_5939_has__derivative__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G5: fun(A,B),F4: filter(A),Xb: B] :
          ( has_derivative(A,B,G,G5,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_rl(fun(A,B),fun(B,fun(A,B)),G),Xb),aa(B,fun(A,B),aTP_Lamp_rl(fun(A,B),fun(B,fun(A,B)),G5),Xb),F4) ) ) ).

% has_derivative_mult_right
tff(fact_5940_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ finite_finite(A,set_or5935395276787703475ssThan(A,A3,B2)) ) ) ).

% infinite_Ioo
tff(fact_5941_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_rn(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),Xb),G),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_mult
tff(fact_5942_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_5943_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,A,aTP_Lamp_pv(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,Xb))),Db)),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_cosh
tff(fact_5944_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,A,aTP_Lamp_pu(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,Xb))),Db)),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_sinh
tff(fact_5945_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_5946_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_5947_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ro(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_rp(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),Xb),G),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_5948_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_rq(A,fun(A,A),Xb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_inverse'
tff(fact_5949_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),Xb: B,F6: fun(B,A),S3: set(B)] :
          ( ( aa(B,A,F2,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F6,topolo174197925503356063within(B,Xb,S3))
           => has_derivative(B,A,aTP_Lamp_rr(fun(B,A),fun(B,A),F2),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_rs(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),Xb),F6),topolo174197925503356063within(B,Xb,S3)) ) ) ) ).

% has_derivative_inverse
tff(fact_5950_DERIV__isconst3,axiom,
    ! [A3: real,B2: real,Xb: real,Y: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( member(real,Xb,set_or5935395276787703475ssThan(real,A3,B2))
       => ( member(real,Y,set_or5935395276787703475ssThan(real,A3,B2))
         => ( ! [X4: real] :
                ( member(real,X4,set_or5935395276787703475ssThan(real,A3,B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) )
           => ( aa(real,real,F2,Xb) = aa(real,real,F2,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_5951_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),Nb: nat] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_rt(fun(A,B),fun(nat,fun(A,B)),F2),Nb),aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_ru(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F6),Xb),Nb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_power
tff(fact_5952_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_rv(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_rw(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_ln
tff(fact_5953_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_ry(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),Xb),G),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_divide
tff(fact_5954_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F6: fun(A,fun(B,C)),Xb: B,S3: set(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,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_sa(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_sc(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I5),F2),F6),Xb),topolo174197925503356063within(B,Xb,S3)) ) ) ).

% has_derivative_prod
tff(fact_5955_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,X6: set(A),F2: fun(A,real),F6: fun(A,real)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,X6))
         => ( has_derivative(A,real,F2,F6,topolo174197925503356063within(A,Xb,X6))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
             => ( member(A,Xb,X6)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sd(fun(A,real),fun(fun(A,real),fun(A,real)),G),F2),aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_se(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G5),Xb),F2),F6),topolo174197925503356063within(A,Xb,X6)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_5956_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_sf(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_5957_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,S: set(A)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,real,aTP_Lamp_sh(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_si(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G5),Xb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_arctan
tff(fact_5958_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( ( cos(real,aa(A,real,G,Xb)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_sj(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_tan
tff(fact_5959_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_sl(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sm(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_5960_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_5961_mlex__eq,axiom,
    ! [A: $tType,F2: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F2,R2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_sn(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F2),R2))) ).

% mlex_eq
tff(fact_5962_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X: product_prod(A,B)] :
            ( member(product_prod(A,B),X,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
           => aa(product_prod(A,B),$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_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),X) ) ) ) ).

% listrel_iff_zip
tff(fact_5963_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_5964_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_5965_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R: 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,R))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel_eq_len
tff(fact_5966_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N4: nat] :
            ( aa(nat,$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)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_5967_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,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_5968_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_5969_mlex__leq,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
     => ( 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(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),mlex_prod(A,F2,R2)) ) ) ).

% mlex_leq
tff(fact_5970_mlex__iff,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat),R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),mlex_prod(A,F2,R2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
        | ( ( aa(A,nat,F2,Xb) = aa(A,nat,F2,Y) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),R2) ) ) ) ).

% mlex_iff
tff(fact_5971_mlex__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),mlex_prod(A,F2,R2)) ) ).

% mlex_less
tff(fact_5972_in__measure,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),measure(A,F2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y)) ) ).

% in_measure
tff(fact_5973_in__finite__psubset,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A4),B4),finite_psubset(A))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
        & finite_finite(A,B4) ) ) ).

% in_finite_psubset
tff(fact_5974_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_so(set(A),fun(set(A),$o)))) ).

% finite_psubset_def
tff(fact_5975_of__char__of,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [A3: A] : aa(char,A,comm_s6883823935334413003f_char(A),aa(A,char,unique5772411509450598832har_of(A),A3)) = modulo_modulo(A,A3,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_5976_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_5977_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [C2: char] : modulo_modulo(A,aa(char,A,comm_s6883823935334413003f_char(A),C2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ).

% of_char_mod_256
tff(fact_5978_nat__of__char__less__256,axiom,
    ! [C2: char] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ).

% nat_of_char_less_256
tff(fact_5979_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_5980_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A,C2: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),Nb) = C2 )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))),Nb) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ) ) ).

% char_of_eq_iff
tff(fact_5981_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_5982_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: $o] : aa(char,A,comm_s6883823935334413003f_char(A),char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($o,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B72)),nil($o)))))))))) ) ).

% of_char_Char
tff(fact_5983_nth__Cons__Suc,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(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_5984_nth__Cons__0,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),zero_zero(nat)) = Xb ).

% nth_Cons_0
tff(fact_5985_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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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_5986_sum__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xb: A,Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ).

% sum_list.Cons
tff(fact_5987_Cons__listrel1__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A),R: 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(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(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),Xb),Y),R)
          & ( 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,R)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_5988_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: A,Xb: B,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_5989_lexord__cons__cons,axiom,
    ! [A: $tType,A3: A,Xb: list(A),B2: A,Y: list(A),R: 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(A,fun(list(A),list(A)),cons(A),A3),Xb)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)),lexord(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),A3),B2),R)
        | ( ( A3 = 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,R)) ) ) ) ).

% lexord_cons_cons
tff(fact_5990_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : enumerate(A,Nb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(product_prod(nat,A),fun(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_5991_nth__Cons__numeral,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(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_5992_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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),list(A),aa(A,fun(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_5993_Cons__in__lex,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A),R: 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(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),lex(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),Xb),Y),R)
          & ( 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,R)) ) ) ) ).

% Cons_in_lex
tff(fact_5994_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),aa(A,fun(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_5995_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_5996_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),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys) ) ) ).

% impossible_Cons
tff(fact_5997_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),Y4: 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),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2)),aa(list(D),list(D),aa(D,fun(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_5998_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),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X4: A,Xs2: list(A),Y4: B,Ys3: list(B),Z3: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$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),$o)),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2)) ) ) )
           => aa(list(C),$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_5999_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),Y4: B,Ys3: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ys3)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_6000_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [Y3: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_6001_Suc__length__conv,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,Nb) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y3: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_6002_length__Cons,axiom,
    ! [A: $tType,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)),cons(A),Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_6003_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)),aa(product_prod(A,B),fun(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),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
           => ! [Y4: B,Ys5: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys5) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) )
                 => ( Xys != zip(A,B,Xs4,Ys5) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6004_find_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      find(A,P,aa(list(A),list(A),aa(A,fun(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_6005_nth__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = case_nat(A,Xb,nth(A,Xs),Nb) ).

% nth_Cons
tff(fact_6006_Suc__le__length__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_6007_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) ) ).

% sorted1
tff(fact_6008_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))
        <=> ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_6009_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))
        <=> ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X) )
            & sorted_wrt(A,ord_less(A),Ys) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_6010_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R: 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),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R))
     => ( ! [X4: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(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),R) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(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,R)) ) ) ) ).

% Cons_listrel1E2
tff(fact_6011_Cons__listrel1E1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),R: 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(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Ys),listrel1(A,R))
     => ( ! [Y4: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),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),Y4),R) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(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,R)) ) ) ) ).

% Cons_listrel1E1
tff(fact_6012_listrel1I1,axiom,
    ! [A: $tType,Xb: A,Y: A,R: 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),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(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)),listrel1(A,R)) ) ).

% listrel1I1
tff(fact_6013_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,R: 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),R)
     => ( 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,R))
       => 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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)),listrel(A,B,R)) ) ) ).

% listrel.Cons
tff(fact_6014_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R: 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),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),Xs),listrel(A,B,R))
     => ~ ! [Y4: B,Ys3: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),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),Y4),R)
             => ~ 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,R)) ) ) ) ).

% listrel_Cons1
tff(fact_6015_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R: 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),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)),listrel(A,B,R))
     => ~ ! [X4: A,Xs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(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),R)
             => ~ 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,R)) ) ) ) ).

% listrel_Cons2
tff(fact_6016_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),aa(A,fun(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_6017_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),aa(A,fun(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_6018_nth__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(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_6019_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R: 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,R))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X4: A,Y4: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),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),Y4),R)
                   => ~ 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,R)) ) ) ) ) ) ).

% listrel.cases
tff(fact_6020_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R: 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,R))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X: A,Y3: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys4) )
            & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3),R)
            & 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,R)) ) ) ) ).

% listrel.simps
tff(fact_6021_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X21: A,X222: list(A)] : size_list(A,Xb,aa(list(A),list(A),aa(A,fun(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)),size_list(A,Xb,X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6022_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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6023_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A),Nb: nat] :
      ( ( Xb != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6024_take__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      take(A,Nb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(Nb = zero_zero(nat),nil(A),aa(list(A),list(A),aa(A,fun(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_6025_Cons__replicate__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = replicate(A,Nb,Y) )
    <=> ( ( Xb = Y )
        & aa(nat,$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_6026_Cons__lenlex__iff,axiom,
    ! [A: $tType,Mb: A,Ms: list(A),Nb: A,Ns: list(A),R: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Mb),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Nb),Ns)),lenlex(A,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Mb),Nb),R) )
        | ( ( Mb = Nb )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6027_String_Ochar__of__ascii__of,axiom,
    ! [C2: char] : aa(char,nat,comm_s6883823935334413003f_char(nat),ascii_of(C2)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)) ).

% String.char_of_ascii_of
tff(fact_6028_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% concat_inth
tff(fact_6029_append__eq__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Us: list(A),Vs: list(A)] :
      ( ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        | ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
     => ( ( append(A,Xs,Us) = append(A,Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_6030_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),append(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_append
tff(fact_6031_sum__list__append,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A),Ys: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),append(A,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),aa(list(A),A,groups8242544230860333062m_list(A),Ys)) ) ).

% sum_list_append
tff(fact_6032_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,append(A,Xs,Ys),append(B,Us,Vs)) = append(product_prod(A,B),zip(A,B,Xs,Us),zip(A,B,Ys,Vs)) ) ) ).

% zip_append
tff(fact_6033_size__list__append,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),Ys: list(A)] : size_list(A,F2,append(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(A,F2,Xs)),size_list(A,F2,Ys)) ).

% size_list_append
tff(fact_6034_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% nth_append_length
tff(fact_6035_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] : aa(nat,A,nth(A,append(A,Xs,Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb)) = aa(nat,A,nth(A,Ys),Nb) ).

% nth_append_length_plus
tff(fact_6036_take__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : take(A,Nb,append(A,Xs,Ys)) = 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_6037_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A] : list_update(A,append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys)),aa(list(A),nat,size_size(list(A)),Xs),Y) = append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ).

% list_update_length
tff(fact_6038_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),aa(nat,fun(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_6039_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)) = append(nat,upt(I,J),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_6040_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),append(nat,upt(I,J),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_6041_rotate__append,axiom,
    ! [A: $tType,L: list(A),Q2: list(A)] : aa(list(A),list(A),rotate(A,aa(list(A),nat,size_size(list(A)),L)),append(A,L,Q2)) = append(A,Q2,L) ).

% rotate_append
tff(fact_6042_enumerate__append__eq,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : enumerate(A,Nb,append(A,Xs,Ys)) = append(product_prod(nat,A),enumerate(A,Nb,Xs),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_6043_replicate__add,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xb: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb),Xb) = append(A,replicate(A,Nb,Xb),replicate(A,Mb,Xb)) ).

% replicate_add
tff(fact_6044_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),Y4: A,Ys5: list(A)] :
            ( ( X4 != Y4 )
            & ( Xs = append(A,Pre,append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)),Xs4)) )
            & ( Ys = append(A,Pre,append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A)),Ys5)) ) ) ) ) ).

% same_length_different
tff(fact_6045_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),append(A,Xs,Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
               => ! [Xa2: A] :
                    ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys))
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2) ) ) ) ) ) ).

% sorted_append
tff(fact_6046_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,append(A,Xs,Ys),I,Xb) = append(A,list_update(A,Xs,I,Xb),Ys) ) ) ).

% list_update_append1
tff(fact_6047_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: 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),K)) = append(nat,upt(I,J),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_6048_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),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_6049_lexord__append__leftD,axiom,
    ! [A: $tType,Xb: list(A),U: list(A),V: list(A),R: 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)),append(A,Xb,U)),append(A,Xb,V)),lexord(A,R))
     => ( ! [A5: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5),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)),U),V),lexord(A,R)) ) ) ).

% lexord_append_leftD
tff(fact_6050_map__of__Cons__code_I2_J,axiom,
    ! [A: $tType,B: $tType,L: B,V: A,Ps: list(product_prod(B,A)),K: B] :
      aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(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)),K) = $ite(L = K,aa(A,option(A),some(A),V),aa(B,option(A),map_of(B,A,Ps),K)) ).

% map_of_Cons_code(2)
tff(fact_6051_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R: 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)),append(A,Xs,Zs)),append(A,Ys,Qs)),lexord(A,R))
     => ( ( Xs != Ys )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
         => ( ( aa(list(A),nat,size_size(list(A)),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,R)) ) ) ) ) ).

% lexord_sufE
tff(fact_6052_lex__append__leftD,axiom,
    ! [A: $tType,R: 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),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)),append(A,Xs,Ys)),append(A,Xs,Zs)),lex(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)),Ys),Zs),lex(A,R)) ) ) ).

% lex_append_leftD
tff(fact_6053_lex__append__left__iff,axiom,
    ! [A: $tType,R: 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),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)),append(A,Xs,Ys)),append(A,Xs,Zs)),lex(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)),Ys),Zs),lex(A,R)) ) ) ).

% lex_append_left_iff
tff(fact_6054_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: 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,R))
     => ( ( 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)),append(A,Xs,Us)),append(A,Ys,Vs)),lex(A,R)) ) ) ).

% lex_append_rightI
tff(fact_6055_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R2: 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,R2))
     => ( ( 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)),append(A,Us,Vs)),append(A,Xs,Ys)),lenlex(A,R2)) ) ) ).

% lenlex_append1
tff(fact_6056_nths__append,axiom,
    ! [A: $tType,L: list(A),L2: list(A),A4: set(nat)] : nths(A,append(A,L,L2),A4) = append(A,nths(A,L,A4),nths(A,L2,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_sp(list(A),fun(set(nat),fun(nat,$o)),L),A4)))) ).

% nths_append
tff(fact_6057_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [Y3: A,Ys4: list(A)] :
          ( ( Xs = append(A,Ys4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% length_Suc_conv_rev
tff(fact_6058_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,aa(list(A),list(A),aa(A,fun(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_6059_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] :
      aa(nat,A,nth(A,append(A,Xs,Ys)),Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,A,nth(A,Xs),Nb),aa(nat,A,nth(A,Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)))) ).

% nth_append
tff(fact_6060_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( append(A,Xs,Ys) = append(A,Ys,Xs) )
         => ? [N: nat,Zs2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N)
              & ( concat(A,replicate(list(A),N,Zs2)) = append(A,Xs,Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_6061_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat,Xb: A] :
      list_update(A,append(A,Xs,Ys),Nb,Xb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),append(A,list_update(A,Xs,Nb,Xb),Ys),append(A,Xs,list_update(A,Ys,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Xb))) ).

% list_update_append
tff(fact_6062_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: 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,R))
     => ~ ! [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),X4),Y4),R)
           => ! [Us2: list(A),Vs2: list(A)] :
                ( ( Xs = append(A,Us2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Vs2)) )
               => ( Ys != append(A,Us2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_6063_listrel1I,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),R)
     => ( ( Xs = append(A,Us,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Vs)) )
       => ( ( Ys = append(A,Us,aa(list(A),list(A),aa(A,fun(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,R)) ) ) ) ).

% listrel1I
tff(fact_6064_lexord__append__left__rightI,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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),A3),B2),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)),append(A,U,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Xb))),append(A,U,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y))),lexord(A,R)) ) ).

% lexord_append_left_rightI
tff(fact_6065_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R: 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)),append(A,Xs,Ys)),append(A,Xs,Zs)),lexord(A,R))
    <=> ( ? [X: A] :
            ( member(A,X,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),X),X),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)),Ys),Zs),lexord(A,R)) ) ) ).

% lexord_same_pref_iff
tff(fact_6066_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W: list(A),R: set(product_prod(A,A)),V: list(A),Z: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W),lexord(A,R))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W)),aa(list(A),nat,size_size(list(A)),U))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,U,V)),append(A,W,Z)),lexord(A,R)) ) ) ).

% lexord_sufI
tff(fact_6067_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(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_6068_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),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)),nil(nat)) ).

% upt_rec
tff(fact_6069_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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys) = append(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_6070_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)
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_6071_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A,R: 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)),append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)))),append(A,Ys,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))),listrel1(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),listrel1(A,R))
          & ( 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),R) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_6072_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),append(B,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),Ys))) ) ).

% horner_sum_append
tff(fact_6073_listrel1__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : listrel1(A,R) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_sq(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R))) ).

% listrel1_def
tff(fact_6074_lexord__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lexord(A,R) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_sr(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R))) ).

% lexord_def
tff(fact_6075_lex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_ss(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R))) ).

% lex_conv
tff(fact_6076_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) = append(A,take(A,I,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_6077_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,append(A,take(A,Nb,Xs),append(A,aa(list(A),list(A),aa(A,fun(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_6078_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)),append(A,take(A,Nb,Xs),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)),drop(A,aa(nat,nat,suc,Nb),Xs)))) ) ) ).

% pos_n_replace
tff(fact_6079_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_6080_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_6081_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_6082_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_6083_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_6084_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_6085_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_6086_drop__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : drop(A,Nb,append(A,Xs,Ys)) = 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_6087_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),aa(A,fun(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_6088_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_6089_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),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
     => ( aa(nat,A,nth(A,Xs),Nb) = Y ) ) ).

% nth_via_drop
tff(fact_6090_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_6091_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_6092_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_6093_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_6094_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) = append(A,take(A,I,Xs),take(A,J,drop(A,I,Xs))) ).

% take_add
tff(fact_6095_append__eq__conv__conj,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: 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_6096_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_6097_nths__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),I5: set(nat)] : nths(A,drop(A,Nb,Xs),I5) = nths(A,Xs,image(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),I5)) ).

% nths_drop
tff(fact_6098_drop__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      drop(A,Nb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(Nb = zero_zero(nat),aa(list(A),list(A),aa(A,fun(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_6099_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( append(A,Xs_1,Xs_2) = 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 = 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 )
          & ( 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_6100_zip__append1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(A),Zs: list(B)] : zip(A,B,append(A,Xs,Ys),Zs) = 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_6101_zip__append2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(B)] : zip(A,B,Xs,append(B,Ys,Zs)) = 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_6102_drop__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,Nb,rev(A,Xs)) = 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_6103_rev__drop,axiom,
    ! [A: $tType,I: nat,Xs: 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),rev(A,Xs)) ).

% rev_drop
tff(fact_6104_rev__take,axiom,
    ! [A: $tType,I: nat,Xs: 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),rev(A,Xs)) ).

% rev_take
tff(fact_6105_take__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : take(A,Nb,rev(A,Xs)) = 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_6106_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),aa(A,fun(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_6107_rotate__drop__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Nb),Xs) = 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_6108_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 = append(A,take(A,I,Xs),aa(list(A),list(A),aa(A,fun(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_6109_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A),A3: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,Xs,I,A3) = append(A,take(A,I,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_6110_lexn__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Nb: nat] : lexn(A,R,Nb) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_st(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R),Nb))) ).

% lexn_conv
tff(fact_6111_list__encode_Oelims,axiom,
    ! [Xb: list(nat),Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( ( ( Xb = nil(nat) )
         => ( Y != zero_zero(nat) ) )
       => ~ ! [X4: nat,Xs2: list(nat)] :
              ( ( Xb = aa(list(nat),list(nat),aa(nat,fun(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),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_6112_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: 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),lexn(A,R,Nb))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = Nb ) ) ) ).

% lexn_length
tff(fact_6113_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list(nat)] : nat_list_encode(aa(list(nat),list(nat),aa(nat,fun(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),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_6114_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),aa(int,fun(list(int),list(int)),cons(int),J),Js))) ).

% upto_aux_rec
tff(fact_6115_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),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_sv(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs),Xys))) ).

% set_relcomp
tff(fact_6116_relcomp_Ocases,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R: set(product_prod(A,C)),S: 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,R,S))
     => ~ ! [B3: 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),B3),R)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B3),A22),S) ) ) ).

% relcomp.cases
tff(fact_6117_relcomp_Osimps,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R: set(product_prod(A,C)),S: 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,R,S))
    <=> ? [A6: A,B5: 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),B5),R)
          & member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B5),C4),S) ) ) ).

% relcomp.simps
tff(fact_6118_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(A,B)),C2: C,S: 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),A3),B2),R)
     => ( member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C2),S)
       => member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C2),relcomp(A,B,C,R,S)) ) ) ).

% relcomp.relcompI
tff(fact_6119_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R: set(product_prod(A,C)),S: set(product_prod(C,B))] :
      ( member(product_prod(A,B),Xz,relcomp(A,C,B,R,S))
     => ~ ! [X4: A,Y4: 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),Y4),R)
             => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y4),Z3),S) ) ) ) ).

% relcompE
tff(fact_6120_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R: set(product_prod(A,C)),S: 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),A3),C2),relcomp(A,C,B,R,S))
     => ~ ! [B3: C] :
            ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B3),R)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B3),C2),S) ) ) ).

% relcompEpair
tff(fact_6121_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: set(product_prod(A,C)),S: set(product_prod(C,B))] : relcomp(A,C,B,R,S) = aa(fun(product_prod(A,B),$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_sw(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),R),S))) ).

% relcomp_unfold
tff(fact_6122_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),aa(int,fun(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_6123_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),aa(int,fun(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_6124_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_6125_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_6126_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_6127_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),aa(int,fun(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_6128_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),aa(int,fun(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_6129_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_6130_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))
     => ( append(A,take(A,Nb,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,drop(A,Nb,Xs))),nil(A))) = take(A,aa(nat,nat,suc,Nb),Xs) ) ) ).

% take_hd_drop
tff(fact_6131_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ql(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sy(fun(nat,A),fun(A,fun(A,A)),C2),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_6132_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( hd(nat,upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_6133_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)
     => ( hd(A,take(A,J,Xs)) = hd(A,Xs) ) ) ).

% hd_take
tff(fact_6134_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sz(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_6135_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ta(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_6136_power__tendsto__0__iff,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tb(nat,fun(fun(A,real),fun(A,real)),Nb),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_6137_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(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),hd(A,Xs)),hd(B,Ys)) ) ) ) ).

% hd_zip
tff(fact_6138_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A3: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_tc(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_6139_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_td(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_6140_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_td(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_6141_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_td(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_6142_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_te(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivative_iff
tff(fact_6143_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_te(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivativeD
tff(fact_6144_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),one_one(real))
         => filterlim(A,real,aTP_Lamp_tf(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A3)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_6145_tendsto__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tg(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),L),C2)),F4) ) ) ).

% tendsto_mult_right
tff(fact_6146_tendsto__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_th(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),L)),F4) ) ) ).

% tendsto_mult_left
tff(fact_6147_tendsto__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),A3: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ti(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)),F4) ) ) ) ).

% tendsto_mult
tff(fact_6148_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F4) ) ) ) ).

% tendsto_mult_one
tff(fact_6149_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_6150_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tl(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_6151_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tm(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_6152_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),A3: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_6153_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_to(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_6154_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
       => filterlim(A,real,aTP_Lamp_tp(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A3)),F4) ) ) ).

% tendsto_arcosh
tff(fact_6155_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_6156_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_tr(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I2),topolo7230453075368039082e_nhds(C,one_one(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_ts(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F4) ) ) ).

% tendsto_one_prod'
tff(fact_6157_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,F2: fun(A,B),D2: B,F4: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tt(B,fun(fun(A,B),fun(A,B)),C2),F2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),D2)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,D2),F4) ) ) ).

% tendsto_add_const_iff
tff(fact_6158_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),A3: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A3),B2)),F4) ) ) ) ).

% tendsto_add
tff(fact_6159_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),A3: B,F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F4)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_tu(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A3),Nb)),F4) ) ) ).

% tendsto_power
tff(fact_6160_tendsto__power__strong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),A3: B,F4: filter(A),G: fun(A,nat),B2: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F4)
         => ( filterlim(A,nat,G,topolo7230453075368039082e_nhds(nat,B2),F4)
           => filterlim(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_tv(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A3),B2)),F4) ) ) ) ).

% tendsto_power_strong
tff(fact_6161_tendsto__real__sqrt,axiom,
    ! [A: $tType,F2: fun(A,real),Xb: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xb),F4)
     => filterlim(A,real,aTP_Lamp_tw(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,Xb)),F4) ) ).

% tendsto_real_sqrt
tff(fact_6162_tendsto__real__root,axiom,
    ! [A: $tType,F2: fun(A,real),Xb: real,F4: filter(A),Nb: nat] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xb),F4)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_tx(fun(A,real),fun(nat,fun(A,real)),F2),Nb),topolo7230453075368039082e_nhds(real,aa(real,real,root(Nb),Xb)),F4) ) ).

% tendsto_real_root
tff(fact_6163_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_ty(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_6164_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
         => ( ( A3 != one_one(real) )
           => ( aa(real,$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_tz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A3),B2)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_6165_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),A3: B,F4: filter(A),G: fun(A,C),B2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F4)
         => ( filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B2),F4)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ua(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),topolo7230453075368039082e_nhds(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)),F4) ) ) ) ).

% tendsto_Pair
tff(fact_6166_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A3: B,F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A3)
         => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A3,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ub(B,fun(fun(B,C),fun(B,C)),A3),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_6167_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R2: real,A3: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( ! [X4: A] :
                ( ( X4 != A3 )
               => ( aa(real,$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),A3))),R2)
                 => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_6168_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X: A] :
                      ( ( ( X != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),S6) )
                     => aa(real,$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,F2,X)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_6169_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A3))),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X4)),L5))),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_6170_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A3: A,R: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [S2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S2)
                & ! [X5: A] :
                    ( ( ( X5 != A3 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),A3))),S2) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X5)),L5))),R) ) ) ) ) ) ).

% LIM_D
tff(fact_6171_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A3: A,D5: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_uc(fun(A,A),fun(A,fun(A,A)),F2),A3),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ud(fun(A,A),fun(A,fun(A,A)),F2),A3),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_6172_LIM__fun__less__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X5))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X5)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_6173_LIM__fun__not__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X5))),R3) )
               => ( aa(real,real,F2,X5) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_6174_LIM__fun__gt__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X5))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F2,X5)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_6175_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A3: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$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),A3))),D6) )
                     => ( aa(A,B,F2,X4) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ue(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_6176_hd__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( hd(A,Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_6177_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_uf(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_6178_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_uf(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_6179_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_ug(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_6180_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( ! [H3: A] :
                ( ( H3 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,H3))),aa(real,real,aa(real,fun(real,real),times_times(real),K5),real_V7770717601297561774m_norm(A,H3))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_6181_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D5),topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_uf(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_6182_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),A3: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_uh(fun(A,B),fun(A,fun(A,B)),F2),A3),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_6183_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))
     => ( hd(A,drop(A,Nb,Xs)) = aa(nat,A,nth(A,Xs),Nb) ) ) ).

% hd_drop_conv_nth
tff(fact_6184_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A3: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X4: A] :
                ( ( X4 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X4)),S)
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),A3),X4),aa(A,A,F2,X4)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_6185_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A3: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X4: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X4)),S)
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),A3),X4),aa(A,A,F2,X4)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_6186_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F2: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( summable(real,F2)
           => ( ! [H3: A,N: nat] :
                  ( ( H3 != zero_zero(A) )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H3),N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N)),real_V7770717601297561774m_norm(A,H3))) ) )
             => filterlim(A,B,aTP_Lamp_ui(fun(A,fun(nat,B)),fun(A,B),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_6187_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_uj(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),top_top(set(real)))) ).

% LIM_cos_div_sin
tff(fact_6188_hd__rotate__conv__nth,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( Xs != nil(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_6189_summable__Leibniz_I2_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(nat,real,A3,zero_zero(nat)))
         => ! [N5: nat] : member(real,suminf(real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),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))),N5))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),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))),N5)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_6190_summable__Leibniz_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,A3,zero_zero(nat))),zero_zero(real))
         => ! [N5: nat] : member(real,suminf(real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),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))),N5)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),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))),N5))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_6191_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A3: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ul(A,fun(fun(nat,A),fun(nat,A)),C2),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_6192_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A3: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_um(A,fun(fun(nat,A),fun(nat,A)),C2),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_6193_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A3: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_un(A,fun(fun(nat,A),fun(nat,A)),C2),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_6194_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,$o),ord_less(A),Y),Xb)
         => ? [U3: fun(nat,A)] :
              ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,U3,N5)),Xb)
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_6195_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,$o),ord_less(A),Xb),Y)
         => ? [U3: fun(nat,A)] :
              ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(nat,A,U3,N5))
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_6196_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),K: nat,A3: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_uo(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A3),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_6197_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),A3: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_uo(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A3),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_6198_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N2: nat,X6: fun(nat,A),Y6: fun(nat,A),Xb: A,Y: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
             => aa(A,$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_6199_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),Xb: A,Y6: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N7: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
                 => aa(A,$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_6200_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,M6: nat,C5: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),C5) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),C5) ) ) ) ).

% Lim_bounded
tff(fact_6201_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,N2: nat,C5: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),aa(nat,A,F2,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),L) ) ) ) ).

% Lim_bounded2
tff(fact_6202_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),Xb: A,A3: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N7: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(nat,A,X6,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xb) ) ) ) ).

% LIMSEQ_le_const
tff(fact_6203_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),Xb: A,A3: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N7: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),A3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A3) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_6204_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A3: A] :
          ( ! [N: nat] : member(A,aa(nat,A,B2,N),S)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,complete_Sup_Sup(A),S)) ) ) ) ).

% Sup_lim
tff(fact_6205_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A3: A] :
          ( ! [N: nat] : member(A,aa(nat,A,B2,N),S)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S)),A3) ) ) ) ).

% Inf_lim
tff(fact_6206_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => filterlim(nat,nat,aTP_Lamp_up(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_6207_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_6208_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_uq(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_6209_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: fun(nat,A),Xb: A] :
          ( topological_monoseq(A,A3)
         => ( filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
           => ( ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,N5)),Xb)
                & ! [M: nat,N5: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,M)),aa(nat,A,A3,N5)) ) )
              | ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(nat,A,A3,N5))
                & ! [M: nat,N5: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,N5)),aa(nat,A,A3,M)) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_6210_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A] : filterlim(nat,A,aTP_Lamp_ur(A,fun(nat,A),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_6211_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_us(fun(nat,A),fun(nat,fun(nat,A)),X6),L),topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_6212_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R3),aa(nat,real,X6,N)) )
     => filterlim(nat,real,aTP_Lamp_ut(fun(nat,real),fun(nat,real),X6),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_6213_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
     => filterlim(nat,real,aTP_Lamp_uu(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_6214_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),L)
       => ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,N5)),E2)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_6215_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_uv(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_6216_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_uw(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_6217_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_ux(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_6218_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_6219_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A3: 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_uy(real,fun(real,fun(nat,real)),Xb),A3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_6220_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_6221_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),C2)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_6222_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_uz(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_6223_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),Xb: real] :
          ( filterlim(nat,real,aTP_Lamp_va(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,Xb),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_6224_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_6225_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N)),L5))),R3) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_6226_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [No3: nat] :
              ! [N5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N5)
               => aa(real,$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,N5)),L5))),R) ) ) ) ) ).

% LIMSEQ_D
tff(fact_6227_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_6228_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim(nat,real,aTP_Lamp_vb(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,exp(real,Xb)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_6229_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F4: filter(A),Xb: B] :
          ( filterlim(A,nat,F2,at_top(nat),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,Xb)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vc(fun(A,nat),fun(B,fun(A,B)),F2),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_6230_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_6231_summable__Leibniz_I1_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => summable(real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)) ) ) ).

% summable_Leibniz(1)
tff(fact_6232_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,S: fun(nat,A),A3: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N: nat] : aa(nat,A,S,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_vd(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),S),topolo7230453075368039082e_nhds(A,A3),at_top(nat))
               => ( Df = A3 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_6233_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_ve(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_6234_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_vf(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_6235_summable,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,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,A3,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => summable(real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)) ) ) ) ).

% summable
tff(fact_6236_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_vg(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K3: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_vh(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K3),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_6237_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_vi(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K3: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_vh(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_6238_summable__Leibniz_I4_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => filterlim(nat,real,aTP_Lamp_vj(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_6239_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_6240_summable__Leibniz_H_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,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,A3,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => filterlim(nat,real,aTP_Lamp_vj(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_6241_summable__Leibniz_H_I2_J,axiom,
    ! [A3: fun(nat,real),Nb: nat] :
      ( filterlim(nat,real,A3,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,A3,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),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_uk(fun(nat,real),fun(nat,real),A3))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_6242_sums__alternating__upper__lower,axiom,
    ! [A3: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N))
       => ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L3: real] :
              ( ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),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))),N5)))),L3)
              & filterlim(nat,real,aTP_Lamp_vj(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L3),at_top(nat))
              & ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L3),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),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))),N5)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_vk(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_6243_summable__Leibniz_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => filterlim(nat,real,aTP_Lamp_vk(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_6244_summable__Leibniz_H_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,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,A3,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => filterlim(nat,real,aTP_Lamp_vk(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_6245_summable__Leibniz_H_I4_J,axiom,
    ! [A3: fun(nat,real),Nb: nat] :
      ( filterlim(nat,real,A3,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,A3,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3)),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_6246_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_vl(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_6247_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_vm(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_Cauchy'
tff(fact_6248_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_vn(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_6249_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_vo(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_6250_bounded__linear__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Xb: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_rl(fun(A,B),fun(B,fun(A,B)),G),Xb)) ) ) ).

% bounded_linear_const_mult
tff(fact_6251_bounded__linear__mult__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Y: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_rk(fun(A,B),fun(B,fun(A,B)),G),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_6252_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_6253_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_vp(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_6254_bounded__linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_6255_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),top_top(A))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),top_top(A))
                & ! [Z2: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Z2)
                   => aa(A,$o,P,Z2) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_6256_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)))
        <=> ? [B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Xb)
              & ! [Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb)
                   => aa(A,$o,P,Y3) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_6257_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)))
          <=> ? [B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Xb)
                & ! [Y3: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Y3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb)
                     => aa(A,$o,P,Y3) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_6258_tendsto__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),G: fun(A,B),Net: filter(A),H: fun(A,B),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_vq(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_vq(fun(A,B),fun(fun(A,B),fun(A,$o)),G),H),Net)
           => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),Net)
             => ( filterlim(A,B,H,topolo7230453075368039082e_nhds(B,C2),Net)
               => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_6259_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F4: filter(A),F2: fun(A,B),Xb: B,G: fun(A,B),Y: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Y),F4)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_vr(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Xb) ) ) ) ) ) ).

% tendsto_le
tff(fact_6260_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_vs(fun(A,B),fun(B,fun(A,$o)),F2),A3),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),Xb) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_6261_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_vt(fun(A,B),fun(B,fun(A,$o)),F2),A3),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),A3) ) ) ) ) ).

% tendsto_upperbound
tff(fact_6262_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
        <=> ( ! [L4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L4),Xb)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_vu(fun(A,B),fun(B,fun(A,$o)),F2),L4),F4) )
            & ! [U4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xb),U4)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_vv(fun(A,B),fun(B,fun(A,$o)),F2),U4),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_6263_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F4: filter(B)] :
          ( ! [A5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_vw(fun(B,A),fun(A,fun(B,$o)),F2),A5),F4) )
         => ( ! [A5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A5)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_vx(fun(B,A),fun(A,fun(B,$o)),F2),A5),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F4) ) ) ) ).

% order_tendstoI
tff(fact_6264_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_vu(fun(A,B),fun(B,fun(A,$o)),F2),A3),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_6265_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),A3)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_vv(fun(A,B),fun(B,fun(A,$o)),F2),A3),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_6266_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B2: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( member(A,X4,set_or5935395276787703475ssThan(A,A3,B2))
             => aa(A,$o,P,X4) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => eventually(A,P,topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_6267_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o),A3: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_vy(fun(A,$o),fun(A,fun(A,$o)),P),A3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_6268_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F2: fun(A,B),F4: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_vz(B,fun(fun(A,B),fun(A,$o)),L),F2),F4)
         => ( ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X4)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_vv(fun(A,B),fun(B,fun(A,$o)),F2),X4),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_6269_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_wa(fun(A,B),fun(B,fun(A,$o)),F2),L),F4)
         => ( ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_vu(fun(A,B),fun(B,fun(A,$o)),F2),X4),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_6270_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [X5: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X5))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X5)),K8)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_6271_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K5: real] :
          ( ! [X4: A,Y4: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4))
         => ( ! [R3: real,X4: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R3),X4)) = aa(B,B,real_V8093663219630862766scaleR(B,R3),aa(A,B,F2,X4))
           => ( ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K5))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_6272_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_vv(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_lessThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_6273_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A3: A] :
          ( ! [X4: A,Y4: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G,X4)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3)))
               => ( ! [B3: A] :
                      ( aa(A,$o,Q,B3)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_6274_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( ( ( A3 != zero_zero(real) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
              & eventually(A,aTP_Lamp_wb(fun(A,real),fun(A,$o),F2),F4) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B2)),F4) ) ) ) ).

% tendsto_powr'
tff(fact_6275_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_wb(fun(A,real),fun(A,$o),F2),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B2)),F4) ) ) ) ) ).

% tendsto_powr2
tff(fact_6276_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_wb(fun(A,real),fun(A,$o),F2),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_6277_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wd(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_6278_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_we(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_6279_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & ? [E4: fun(A,B)] :
                ( ! [H5: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xb)),aa(A,B,F6,H5))),aa(A,B,E4,H5))
                & filterlim(A,real,aTP_Lamp_wf(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_6280_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_vl(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_within
tff(fact_6281_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D5: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D5)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_wg(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D5),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_6282_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_vn(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_6283_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wh(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_6284_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_6285_eventually__at__left__real,axiom,
    ! [B2: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B2),A3)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_wi(real,fun(real,fun(real,$o)),B2),A3),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3))) ) ).

% eventually_at_left_real
tff(fact_6286_filterlim__pow__at__top,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tb(nat,fun(fun(A,real),fun(A,real)),Nb),F2),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_6287_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_6288_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wk(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_6289_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,F2: fun(A,real),F4: filter(A)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),S),zero_zero(real))
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wl(real,fun(fun(A,real),fun(A,real)),S),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_neg_powr
tff(fact_6290_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => ( eventually(A,aTP_Lamp_wm(fun(A,real),fun(A,$o),G),F4)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wn(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_6291_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_wm(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,aTP_Lamp_wo(fun(A,real),fun(A,real),F2),at_top(real),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_6292_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_wm(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_wo(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ) ).

% filterlim_inverse_at_top
tff(fact_6293_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_wp(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_6294_filterlim__tan__at__left,axiom,
    filterlim(real,real,tan(real),at_top(real),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),set_ord_lessThan(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% filterlim_tan_at_left
tff(fact_6295_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),X4)
         => ? [Y5: real] :
              ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X4,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),zero_zero(real)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_6296_tendsto__arctan__at__top,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),at_top(real)) ).

% tendsto_arctan_at_top
tff(fact_6297_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X4)
             => aa(A,$o,P,X4) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_6298_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N6: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N6),N4)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_top_linorder
tff(fact_6299_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N6: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N6),N4)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_top_dense
tff(fact_6300_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,$o)] :
      ( ! [X4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),X4)
         => aa(nat,$o,P,X4) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_6301_eventually__sequentially,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N6: nat] :
        ! [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N4)
         => aa(nat,$o,P,N4) ) ) ).

% eventually_sequentially
tff(fact_6302_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less_eq(A),C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_6303_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_6304_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))
    <=> ! [N6: nat] : eventually(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),F4) ) ).

% le_sequentially
tff(fact_6305_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A)] :
          ( ! [X4: A,Y4: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G,X4)) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_6306_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_wq(fun(A,$o),fun(A,$o),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_6307_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_top
tff(fact_6308_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_6309_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,at_top(B),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ws(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
           => filterlim(A,B,G,at_top(B),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_6310_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_wt(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_6311_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,Nb: nat,B4: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_wu(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C2),Nb),B4),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_6312_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_wv(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_bounded_partials
tff(fact_6313_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_6314_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_6315_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_6316_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_6317_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ finite_finite(A,set_or3652927894154168847AtMost(A,A3,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% infinite_Ioc_iff
tff(fact_6318_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C2: A,A3: A,B2: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),C2),set_or3652927894154168847AtMost(A,A3,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ).

% image_add_greaterThanAtMost
tff(fact_6319_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_6320_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_6321_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_6322_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_6323_less__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F8: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less(filter(A)),F4),F8)
    <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F8)
        & ~ aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F8),F4) ) ) ).

% less_filter_def
tff(fact_6324_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_6325_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ finite_finite(A,set_or3652927894154168847AtMost(A,A3,B2)) ) ) ).

% infinite_Ioc
tff(fact_6326_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( ( set_or3652927894154168847AtMost(A,A3,B2) = set_or3652927894154168847AtMost(A,C2,D2) )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2) )
            | ( ( A3 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_6327_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.head
tff(fact_6328_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.head
tff(fact_6329_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_6330_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_6331_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_6332_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ww(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_6333_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ww(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_6334_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)
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_6335_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_wm(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,F2,at_top(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_6336_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( ( C2 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F4)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_6337_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wy(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_6338_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_wz(fun(A,B),fun(nat,fun(A,B)),F2),Nb),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_6339_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_infinity(A))
        <=> ? [B5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B5)
              & ! [X: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B5),real_V7770717601297561774m_norm(A,X))
                 => aa(A,$o,P2,X) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_6340_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,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_6341_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F4: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_6342_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C2: real,F2: fun(A,B),F4: filter(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
         => ( filterlim(A,B,F2,at_infinity(B),F4)
          <=> ! [R5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),R5)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_xa(fun(A,B),fun(real,fun(A,$o)),F2),R5),F4) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_6343_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_6344_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_xb(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_6345_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xb: A,S3: set(A),F2: fun(A,B),F6: fun(A,B)] :
          ( member(A,Xb,S3)
         => ( topolo1002775350975398744n_open(A,S3)
           => ( has_derivative(A,B,F2,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,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xb)),aa(A,B,F6,H5))),aa(A,B,E4,H5)) ) )
                    & filterlim(A,real,aTP_Lamp_wf(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_6346_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E: real,F6: fun(A,B),S: set(A),Xb: A,F2: fun(A,B),H6: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => ( real_V3181309239436604168linear(A,B,F6)
           => ( ! [Y4: A] :
                  ( member(A,Y4,S)
                 => ( ( Y4 != Xb )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y4,Xb)),E)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,Y4)),aa(A,B,F2,Xb))),aa(A,B,F6,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y4),Xb))))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y4),Xb)))),aa(A,real,H6,Y4)) ) ) )
             => ( filterlim(A,real,H6,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S))
               => has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_6347_dist__add__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A,C2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel
tff(fact_6348_dist__add__cancel2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [B2: A,A3: A,C2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel2
tff(fact_6349_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Y))
        <=> ( Xb != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_6350_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_xc(product_prod(A,A),$o))) ) ).

% open_diagonal_complement
tff(fact_6351_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Z: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xb,Z)),real_V557655796197034286t_dist(A,Y,Z))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Y)),E) ) ) ).

% dist_triangle_lt
tff(fact_6352_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Y: A,E1: real,X22: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,Y)),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X22)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_6353_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Y)) ) ) ).

% dist_pos_lt
tff(fact_6354_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Y)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_6355_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,D2: real] : topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_xd(A,fun(real,fun(A,$o)),Xb),D2))) ) ).

% open_ball
tff(fact_6356_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,Xb: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,Xb)),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Y)),E) ) ) ).

% dist_commute_lessI
tff(fact_6357_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S3: set(A)] :
          ( topolo1002775350975398744n_open(A,S3)
        <=> ! [X: A] :
              ( member(A,X,S3)
             => ? [E4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
                  & ! [Y3: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y3,X)),E4)
                     => member(A,Y3,S3) ) ) ) ) ) ).

% open_dist
tff(fact_6358_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A4: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A4)
         => ( ! [X4: A] :
                ( member(A,X4,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xb) )
           => ~ member(A,aa(set(A),A,complete_Sup_Sup(A),A4),A4) ) ) ) ).

% Sup_notin_open
tff(fact_6359_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A4: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A4)
         => ( ! [X4: A] :
                ( member(A,X4,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X4) )
           => ~ member(A,aa(set(A),A,complete_Inf_Inf(A),A4),A4) ) ) ) ).

% Inf_notin_open
tff(fact_6360_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,$o),ord_less(A),Xb),Y)
             => ? [B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B3)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Xb,B3)),S3) ) ) ) ) ) ).

% open_right
tff(fact_6361_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,$o),ord_less(A),Y),Xb)
             => ? [B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),Xb)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B3,Xb)),S3) ) ) ) ) ) ).

% open_left
tff(fact_6362_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_xe(product_prod(A,A),$o))) ) ).

% open_superdiagonal
tff(fact_6363_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_xf(product_prod(A,A),$o))) ) ).

% open_subdiagonal
tff(fact_6364_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A,S3: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,S3))
        <=> ? [D4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
              & ! [X: A] :
                  ( member(A,X,S3)
                 => ( ( ( X != A3 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A3)),D4) )
                   => aa(A,$o,P,X) ) ) ) ) ) ).

% eventually_at
tff(fact_6365_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A3))
        <=> ? [D4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
              & ! [X: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A3)),D4)
                 => aa(A,$o,P,X) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_6366_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,A3: A,S3: set(A),D2: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,A3,S3))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,A3,S3)
             => ( ! [X4: A] :
                    ( member(A,X4,S3)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A3)),D2)
                     => ( aa(A,A,F2,X4) = aa(A,A,G,X4) ) ) )
               => has_field_derivative(A,G,F6,topolo174197925503356063within(A,A3,S3)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_6367_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S: set(A),D2: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,Xb,S)
             => ( ! [X9: A] :
                    ( member(A,X9,S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X9,Xb)),D2)
                     => ( aa(A,B,F2,X9) = aa(A,B,G,X9) ) ) )
               => has_derivative(A,B,G,F6,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_6368_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M9: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M2)
                 => ! [N: nat] :
                      ( aa(nat,$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,M2),aa(nat,A,X6,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% metric_CauchyI
tff(fact_6369_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => ? [M7: nat] :
              ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M)
               => ! [N5: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N5)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M),aa(nat,A,X6,N5))),E) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_6370_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [N6: nat] :
                ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S,N4),aa(nat,A,S,N6))),E4) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_6371_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_6372_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S8: set(A)] :
              ( topolo1002775350975398744n_open(A,S8)
             => ( member(A,F0,S8)
               => ? [N6: nat] :
                  ! [N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N4)
                   => member(A,aa(nat,A,F2,N4),S8) ) ) ) ) ) ).

% lim_explicit
tff(fact_6373_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X1: A,E: real,X22: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X1)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X22)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X22)),E) ) ) ) ).

% dist_triangle_half_r
tff(fact_6374_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Y: A,E: real,X22: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X22)),E) ) ) ) ).

% dist_triangle_half_l
tff(fact_6375_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,Xb: A,S3: set(A),D2: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xb,S3))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( ! [X9: A] :
                  ( member(A,X9,S3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X9,Xb))
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X9,Xb)),D2)
                     => ( aa(A,B,F2,X9) = aa(A,B,G,X9) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% Lim_transform_within
tff(fact_6376_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A,S3: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,S3))
        <=> ? [D4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
              & ! [X: A] :
                  ( member(A,X,S3)
                 => ( ( ( X != A3 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,A3)),D4) )
                   => aa(A,$o,P,X) ) ) ) ) ) ).

% eventually_at_le
tff(fact_6377_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,X22: A,E: real,X32: A,X42: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X22)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,X32)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X42)),E) ) ) ) ) ).

% dist_triangle_third
tff(fact_6378_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G6: filter(B),Xb: A,S3: set(A),F4: filter(B),D2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G6,topolo174197925503356063within(A,Xb,S3))
         => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G6),F4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
             => ( ! [X9: A] :
                    ( member(A,X9,S3)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X9,Xb))
                     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X9,Xb)),D2)
                       => ( aa(A,B,F2,X9) = aa(A,B,G,X9) ) ) ) )
               => filterlim(A,B,F2,F4,topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_6379_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M9: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M2),aa(nat,A,X6,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI'
tff(fact_6380_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M8: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M5)
                 => ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N4)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F2,M5),aa(nat,A,F2,N4))),E4) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_6381_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),E: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_xg(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E),F4) ) ) ) ).

% tendstoD
tff(fact_6382_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_xg(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E2),F4) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% tendstoI
tff(fact_6383_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_xg(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E4),F4) ) ) ) ).

% tendsto_iff
tff(fact_6384_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X: A] :
                      ( ( ( X != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A3)),S6) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X),L5)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_6385_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A3: A,R: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [S2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S2)
                & ! [X5: A] :
                    ( ( ( X5 != A3 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X5,A3)),S2) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X5),L5)),R) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_6386_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A3: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A3)),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L5)),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_6387_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A3: A,R2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ( ! [X4: A] :
                  ( ( X4 != A3 )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A3)),R2)
                   => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_6388_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_6389_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N),L5)),R3) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_6390_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [No3: nat] :
              ! [N5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N5),L5)),R) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_6391_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_6392_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A3: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A3)),D6) )
                     => ( aa(A,B,F2,X4) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xh(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_6393_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A3: B,S3: set(B),F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A3)
         => ( member(B,A3,S3)
           => ( topolo1002775350975398744n_open(B,S3)
             => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A3,S3))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ub(B,fun(fun(B,C),fun(B,C)),A3),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_6394_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_6395_totally__bounded__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S3: set(A)] :
          ( topolo6688025880775521714ounded(A,S3)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [K2: set(A)] :
                  ( finite_finite(A,K2)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(A,set(A),aTP_Lamp_xj(real,fun(A,set(A)),E4),K2))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_6396_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( 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_xk(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_6397_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N6: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N4),N6)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_6398_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N6: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N4),N6)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_bot_dense
tff(fact_6399_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_kw(A,fun(A,$o)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_6400_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_xl(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_6401_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z6),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_xm(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_6402_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_xm(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_bot
tff(fact_6403_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_xn(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_6404_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_bot(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_6405_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_xo(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,F2,at_bot(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_6406_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z6),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_xp(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_6407_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_xo(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_wo(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ) ).

% filterlim_inverse_at_bot
tff(fact_6408_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_6409_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
         => ? [Y5: real] :
              ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X4,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y5) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_6410_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( ~ 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_xk(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_6411_tendsto__arctan__at__bot,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),at_bot(real)) ).

% tendsto_arctan_at_bot
tff(fact_6412_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [Y3: B,K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_xq(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),Y3),K6),F4) ) ) ) ).

% Bfun_metric_def
tff(fact_6413_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & eventually(A,aa(real,fun(A,$o),aTP_Lamp_xr(fun(A,B),fun(real,fun(A,$o)),F2),K6),F4) ) ) ) ).

% Bfun_def
tff(fact_6414_Bseq__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( bfun(nat,A,F2,at_top(nat))
         => ( bfun(nat,A,G,at_top(nat))
           => bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_6415_Bseq__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,F2,at_top(nat))
         => bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_xt(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_6416_Bseq__add__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_xt(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_add_iff
tff(fact_6417_Bseq__offset,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),K: nat] :
          ( bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xu(fun(nat,A),fun(nat,fun(nat,A)),X6),K),at_top(nat))
         => bfun(nat,A,X6,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_6418_Bseq__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),K: nat] :
          ( bfun(nat,A,X6,at_top(nat))
         => bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xu(fun(nat,A),fun(nat,fun(nat,A)),X6),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_6419_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_de(A,fun(fun(nat,A),fun(nat,A)),C2),F2),at_top(nat))
          <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_6420_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N4))),K6) ) ) ) ).

% Bseq_def
tff(fact_6421_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K5: real,X6: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
         => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N))),K5)
           => bfun(nat,A,X6,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_6422_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
         => ~ ! [K8: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
               => ~ ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),K8) ) ) ) ).

% BseqE
tff(fact_6423_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),K8) ) ) ) ).

% BseqD
tff(fact_6424_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N6: nat] :
            ! [N4: nat] : aa(real,$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,N6))) ) ) ).

% Bseq_iff1a
tff(fact_6425_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_6426_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
              & ? [X: A] :
                ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N4)),aa(A,A,uminus_uminus(A),X)))),K2) ) ) ) ).

% Bseq_iff2
tff(fact_6427_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
              & ? [N6: nat] :
                ! [N4: nat] : aa(real,$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,N6))))),K2) ) ) ) ).

% Bseq_iff3
tff(fact_6428_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
         => ~ ! [B9: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B9)
               => ~ eventually(A,aa(real,fun(A,$o),aTP_Lamp_xr(fun(A,B),fun(real,fun(A,$o)),F2),B9),F4) ) ) ) ).

% BfunE
tff(fact_6429_filterlim__tan__at__right,axiom,
    filterlim(real,real,tan(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))) ).

% filterlim_tan_at_right
tff(fact_6430_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F4: filter(A)] :
          ( topolo6773858410816713723filter(A,F4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [P5: fun(A,$o)] :
                  ( eventually(A,P5,F4)
                  & ! [X: A,Y3: A] :
                      ( ( aa(A,$o,P5,X)
                        & aa(A,$o,P5,Y3) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y3)),E4) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_6431_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_greaterThan(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),I) ) ) ).

% greaterThan_iff
tff(fact_6432_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_6433_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_6434_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_6435_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)))
        <=> ? [B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B5)
              & ! [Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),B5)
                   => aa(A,$o,P,Y3) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_6436_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)))
          <=> ? [B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B5)
                & ! [Y3: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),B5)
                     => aa(A,$o,P,Y3) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_6437_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( topolo174197925503356063within(A,A3,set_or1337092689740270186AtMost(A,A3,B2)) = topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3)) ) ) ) ).

% at_within_Icc_at_right
tff(fact_6438_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_6439_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B2: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( member(A,X4,set_or5935395276787703475ssThan(A,A3,B2))
             => aa(A,$o,P,X4) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => eventually(A,P,topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% eventually_at_rightI
tff(fact_6440_eventually__at__right__real,axiom,
    ! [A3: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_wi(real,fun(real,fun(real,$o)),A3),B2),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3))) ) ).

% eventually_at_right_real
tff(fact_6441_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),P2: B,F12: filter(A),C2: B,L: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,P2,set_ord_greaterThan(B,P2)),F12)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => ( ( L = aa(B,B,aa(B,fun(B,B),times_times(B),C2),P2) )
             => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_xv(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo174197925503356063within(B,L,set_ord_greaterThan(B,L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_6442_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_vu(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_greaterThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_6443_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A3: A] :
          ( ! [X4: A,Y4: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G,X4)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3)))
               => ( ! [B3: A] :
                      ( aa(A,$o,Q,B3)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_6444_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S3: set(A)] :
          ( ! [A5: A,B3: A,X4: A] :
              ( member(A,A5,S3)
             => ( member(A,B3,S3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B3)
                   => member(A,X4,S3) ) ) ) )
         => ? [A5: A,B3: A] :
              ( ( S3 = bot_bot(set(A)) )
              | ( S3 = top_top(set(A)) )
              | ( S3 = set_ord_lessThan(A,B3) )
              | ( S3 = set_ord_atMost(A,B3) )
              | ( S3 = set_ord_greaterThan(A,A5) )
              | ( S3 = set_ord_atLeast(A,A5) )
              | ( S3 = set_or5935395276787703475ssThan(A,A5,B3) )
              | ( S3 = set_or3652927894154168847AtMost(A,A5,B3) )
              | ( S3 = set_or7035219750837199246ssThan(A,A5,B3) )
              | ( S3 = set_or1337092689740270186AtMost(A,A5,B3) ) ) ) ) ).

% interval_cases
tff(fact_6445_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( ! [F3: fun(nat,A)] :
                ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(nat,A,F3,N5))
               => ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N5)),B2)
                 => ( order_antimono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_xw(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_6446_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_atLeast(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),I) ) ) ).

% atLeast_iff
tff(fact_6447_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_6448_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),K),set_ord_atLeast(A,I)) = set_ord_atLeast(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),K),I)) ) ).

% image_add_atLeast
tff(fact_6449_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_6450_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( order_antimono(nat,A,F2)
         => ( aa(nat,$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,F2,J)),aa(nat,A,F2,I)) ) ) ) ).

% decseqD
tff(fact_6451_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_6452_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N4))),aa(nat,A,F2,N4)) ) ) ).

% decseq_Suc_iff
tff(fact_6453_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_6454_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A4,aa(nat,nat,suc,I))),aa(nat,A,A4,I)) ) ) ).

% decseq_SucD
tff(fact_6455_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_6456_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y3)),aa(A,B,F2,X)) ) ) ) ).

% antimono_def
tff(fact_6457_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y4)),aa(A,B,F2,X4)) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_6458_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,Xb)) ) ) ) ).

% antimonoE
tff(fact_6459_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,Xb)) ) ) ) ).

% antimonoD
tff(fact_6460_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,A3)),set_ord_greaterThan(A,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_6461_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_6462_tendsto__at__right__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,B2: A,X6: fun(A,B),L5: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( ! [S4: fun(nat,A)] :
                ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(nat,A,S4,N5))
               => ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S4,N5)),B2)
                 => ( order_antimono(nat,A,S4)
                   => ( filterlim(nat,A,S4,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_xx(fun(A,B),fun(fun(nat,A),fun(nat,B)),X6),S4),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X6,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_6463_Real_Opositive_Orep__eq,axiom,
    ! [Xb: real] :
      ( aa(real,$o,positive2,Xb)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K2: nat] :
            ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N4)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,aa(real,fun(nat,rat),rep_real2,Xb),N4)) ) ) ) ).

% Real.positive.rep_eq
tff(fact_6464_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_6465_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_6466_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_6467_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_6468_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_6469_inverse__real__def,axiom,
    inverse_inverse(real) = aa(fun(fun(nat,rat),fun(nat,rat)),fun(real,real),map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real2,real2),aTP_Lamp_xy(fun(nat,rat),fun(nat,rat))) ).

% inverse_real_def
tff(fact_6470_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),$o))),aTP_Lamp_ya(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_6471_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A7: A,B6: B,R: set(product_prod(A,A)),S: 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),A3),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B6)),lex_prod(A,B,R,S))
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A7),R)
        | ( ( A3 = A7 )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B6),S) ) ) ) ).

% in_lex_prod
tff(fact_6472_uminus__real__def,axiom,
    uminus_uminus(real) = aa(fun(fun(nat,rat),fun(nat,rat)),fun(real,real),map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real2,real2),aTP_Lamp_or(fun(nat,rat),fun(nat,rat))) ).

% uminus_real_def
tff(fact_6473_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),R2: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R2) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_yc(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),P),R2)))) ).

% same_fst_def
tff(fact_6474_card__partition,axiom,
    ! [A: $tType,C5: set(set(A)),K: nat] :
      ( finite_finite(set(A),C5)
     => ( finite_finite(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5))
       => ( ! [C3: set(A)] :
              ( member(set(A),C3,C5)
             => ( aa(set(A),nat,finite_card(A),C3) = K ) )
         => ( ! [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),K),aa(set(set(A)),nat,finite_card(set(A)),C5)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)) ) ) ) ) ) ).

% card_partition
tff(fact_6475_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_6476_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% inf.bounded_iff
tff(fact_6477_same__fstI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Xb: A,Y7: B,Y: B,R2: fun(A,set(product_prod(B,B)))] :
      ( aa(A,$o,P,Xb)
     => ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y7),Y),aa(A,set(product_prod(B,B)),R2,Xb))
       => member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y7)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y)),same_fst(A,B,P,R2)) ) ) ).

% same_fstI
tff(fact_6478_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A4: set(A),P: fun(A,$o),F2: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yd(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F2)),A4) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6479_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A4: set(A),F2: fun(A,B),P: fun(A,$o)] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ye(fun(A,B),fun(fun(A,$o),fun(A,B)),F2),P)),A4) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6480_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B4: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ).

% Sup_inter_less_eq
tff(fact_6481_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_6482_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_6483_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_6484_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_6485_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2) ) ) ) ).

% le_infE
tff(fact_6486_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A3)
         => ( aa(A,$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),A3),B2)) ) ) ) ).

% le_infI
tff(fact_6487_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D2)) ) ) ) ).

% inf_mono
tff(fact_6488_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),Xb) ) ) ).

% le_infI1
tff(fact_6489_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xb: A,A3: A] :
          ( aa(A,$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),A3),B2)),Xb) ) ) ).

% le_infI2
tff(fact_6490_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.orderE
tff(fact_6491_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% inf.orderI
tff(fact_6492_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X4: A,Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X4),Y4)),X4)
         => ( ! [X4: A,Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X4),Y4)),Y4)
           => ( ! [X4: A,Y4: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,aa(A,fun(A,A),F2,Y4),Z3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F2,Xb),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_6493_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_6494_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb1
tff(fact_6495_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_6496_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_6497_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_6498_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% inf.boundedE
tff(fact_6499_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ) ) ).

% inf.boundedI
tff(fact_6500_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_6501_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.order_iff
tff(fact_6502_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),A3) ) ).

% inf.cobounded1
tff(fact_6503_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2) ) ).

% inf.cobounded2
tff(fact_6504_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb_iff1
tff(fact_6505_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_6506_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).

% inf.coboundedI1
tff(fact_6507_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).

% inf.coboundedI2
tff(fact_6508_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).

% inf.strict_coboundedI2
tff(fact_6509_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).

% inf.strict_coboundedI1
tff(fact_6510_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_6511_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% inf.strict_boundedE
tff(fact_6512_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_6513_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb3
tff(fact_6514_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xb: A,A3: A] :
          ( aa(A,$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),A3),B2)),Xb) ) ) ).

% less_infI2
tff(fact_6515_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),Xb) ) ) ).

% less_infI1
tff(fact_6516_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),Ta: set(A)] : image(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),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),A3),S)),image(A,A,aa(A,fun(A,A),plus_plus(A),A3),Ta)) ) ).

% translation_Int
tff(fact_6517_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_6518_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D2)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),A3) ) ) ) ).

% Ioc_disjoint
tff(fact_6519_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),B4: set(A)] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_yf(fun(A,B),fun(set(A),fun(A,B)),G),B4)),A4) ) ) ) ).

% prod.inter_restrict
tff(fact_6520_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A,Xb: A] :
          aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,K)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),K),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_6521_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),B4: set(A)] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))) ) ) ) ).

% sum.Int_Diff
tff(fact_6522_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),B4: set(A)] :
          ( finite_finite(A,A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))) ) ) ) ).

% prod.Int_Diff
tff(fact_6523_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite(A,T5)
         => ( 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)),T5),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),T5))
                   => ( aa(A,B,G,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),T5))
                     => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T5) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_6524_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [A5: A,B3: A] :
              ( member(A,Xb,set_ord_lessThan(A,A5))
              & member(A,Y,set_ord_greaterThan(A,B3))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,A5)),set_ord_greaterThan(A,B3)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_6525_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_6526_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( 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_yg(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% sum.If_cases
tff(fact_6527_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( 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_yh(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% prod.If_cases
tff(fact_6528_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A4: set(A),F2: fun(A,B),B2: B] :
          ( finite_finite(A,A4)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(B,fun(A,B),aTP_Lamp_yi(fun(A,B),fun(B,fun(A,B)),F2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_yj(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_yk(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_6529_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(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_6530_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_6531_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: A,B4: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),aa(set(A),A,complete_Inf_Inf(A),image(nat,A,aTP_Lamp_yl(A,fun(nat,A),B4),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B4) ) ).

% INF_nat_binary
tff(fact_6532_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A)] : lattic7752659483105999362nf_fin(A,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_ym(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Inf_fin.eq_fold'
tff(fact_6533_inf__enat__def,axiom,
    inf_inf(extended_enat) = ord_min(extended_enat) ).

% inf_enat_def
tff(fact_6534_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_6535_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B)),X5: A,Xa: B] :
      ( aa(B,$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_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R2),S3)) ) ).

% inf_Int_eq2
tff(fact_6536_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite(A,A4)
         => ( member(A,A3,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A4)),A3) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_6537_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic7752659483105999362nf_fin(A,A4))
             => ! [A9: A] :
                  ( member(A,A9,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A9) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_6538_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A5) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic7752659483105999362nf_fin(A,A4)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_6539_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic7752659483105999362nf_fin(A,A4))
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_6540_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A)] :
          ( ~ finite_finite(A,A4)
         => ( lattic7752659483105999362nf_fin(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_6541_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( finite_finite(A,B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,B4)),lattic7752659483105999362nf_fin(A,A4)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_6542_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_6543_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys) = 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_yn(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Xb),Xs),Ys) ).

% zip_Cons1
tff(fact_6544_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_6545_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = 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_yo(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys),Xs) ).

% zip_Cons
tff(fact_6546_times__real__def,axiom,
    times_times(real) = aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(real,fun(real,real)),map_fun(real,fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),fun(real,real),rep_real2,map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real2,real2)),aTP_Lamp_os(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% times_real_def
tff(fact_6547_plus__real__def,axiom,
    plus_plus(real) = aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(real,fun(real,real)),map_fun(real,fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),fun(real,real),rep_real2,map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real2,real2)),aTP_Lamp_ou(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% plus_real_def
tff(fact_6548_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_qh(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_6549_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A3: A,F2: fun(A,B),C2: fun(nat,B),K5: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_yp(fun(nat,B),fun(B,fun(nat,B)),C2),K5))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,A3))),real_V7770717601297561774m_norm(B,K5))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_yr(fun(A,B),fun(fun(nat,B),fun(A,B)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_6550_continuous__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aa(B,fun(A,B),aTP_Lamp_ys(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_right
tff(fact_6551_continuous__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aa(B,fun(A,B),aTP_Lamp_yt(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_left
tff(fact_6552_continuous__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_yu(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult'
tff(fact_6553_continuous__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_yv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult
tff(fact_6554_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_yw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_6555_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_yx(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_real_sqrt
tff(fact_6556_continuous__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,nat,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,nat),fun(A,B),aTP_Lamp_yy(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_power'
tff(fact_6557_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),Nb: nat] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aa(nat,fun(A,B),aTP_Lamp_yz(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% continuous_power
tff(fact_6558_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),Nb: nat] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aa(nat,fun(A,real),aTP_Lamp_za(fun(A,real),fun(nat,fun(A,real)),F2),Nb)) ) ) ).

% continuous_real_root
tff(fact_6559_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_zb(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% isCont_Pair
tff(fact_6560_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,C,F4,G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F4,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_zb(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_Pair
tff(fact_6561_IVT2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),B2: B,Y: A,A3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,B2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B2)
             => ( ! [X4: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X4,top_top(set(B))),F2) )
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_6562_IVT,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),A3: B,Y: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,A3)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,B2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B2)
             => ( ! [X4: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X4,top_top(set(B))),F2) )
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_6563_isCont__real__sqrt,axiom,
    ! [Xb: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),sqrt) ).

% isCont_real_sqrt
tff(fact_6564_isCont__real__root,axiom,
    ! [Xb: real,Nb: nat] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),root(Nb)) ).

% isCont_real_root
tff(fact_6565_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),G)
           => ( ( aa(A,B,G,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),aa(fun(A,B),fun(A,B),aTP_Lamp_zc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_6566_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_yv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_mult
tff(fact_6567_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_yw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_6568_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A3: A,F2: fun(A,B),Nb: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_yz(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% isCont_power
tff(fact_6569_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M7: A] :
              ! [X5: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X5),B2) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X5)),M7) ) ) ) ) ).

% isCont_bounded
tff(fact_6570_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M7: A] :
                ( ! [X5: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X5)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X5),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X5)),M7) )
                & ? [X4: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
                    & ( aa(real,A,F2,X4) = M7 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_6571_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M7: A] :
                ( ! [X5: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X5)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X5),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M7),aa(real,A,F2,X5)) )
                & ? [X4: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
                    & ( aa(real,A,F2,X4) = M7 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_6572_isCont__inverse__function2,axiom,
    ! [A3: real,Xb: real,B2: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
       => ( ! [Z3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Z3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),B2)
               => ( aa(real,real,G,aa(real,real,F2,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),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))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xb),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_6573_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( ( aa(A,B,G,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_zc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_6574_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A,Xb: A] :
          ( has_field_derivative(A,F2,L,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ? [G7: fun(A,A)] :
              ( ! [Z2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,F2,Z2)),aa(A,A,F2,Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G7,Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),Xb))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),G7)
              & ( aa(A,A,G7,Xb) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_6575_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M7: A] :
                ( ! [X5: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X5)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X5),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X5)),M7) )
                & ! [N7: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N7),M7)
                   => ? [X4: real] :
                        ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X4)
                        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),N7),aa(real,A,F2,X4)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_6576_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A3))
             => ( ( aa(A,real,F2,A3) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,A3))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S),aa(fun(A,real),fun(A,real),aTP_Lamp_zd(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_6577_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_6578_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xb: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ze(A,fun(fun(A,B),fun(A,B)),Xb),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,Xb)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_6579_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D5: real,G: fun(real,real),Xb: real,A3: real,B2: real] :
      ( has_field_derivative(real,F2,D5,topolo174197925503356063within(real,aa(real,real,G,Xb),top_top(set(real))))
     => ( ( D5 != zero_zero(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
           => ( ! [Y4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Y4)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),B2)
                   => ( aa(real,real,F2,aa(real,real,G,Y4)) = Y4 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D5),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_6580_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A,C2: fun(nat,A),Nb: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_zf(fun(nat,A),fun(nat,fun(A,A)),C2),Nb)) ) ).

% isCont_polynom
tff(fact_6581_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_6582_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_6583_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [Y4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C2),Y4))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_qh(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_6584_LIM__less__bound,axiom,
    ! [B2: real,Xb: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B2),Xb)
     => ( ! [X4: real] :
            ( member(real,X4,set_or5935395276787703475ssThan(real,B2,Xb))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X4)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_6585_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A3))
             => ( ( aa(A,real,F2,A3) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,A3))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_zd(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_6586_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_6587_isCont__inverse__function,axiom,
    ! [D2: real,Xb: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
     => ( ! [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))),D2)
           => ( aa(real,real,G,aa(real,real,F2,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))),D2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xb),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_6588_GMVT_H,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),G: fun(real,real),G5: fun(real,real),F6: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( ! [Z3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),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))),F2) ) )
       => ( ! [Z3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),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))),G) ) )
         => ( ! [Z3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
                 => has_field_derivative(real,G,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),A3),Z3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
                   => has_field_derivative(real,F2,aa(real,real,F6,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
             => ? [C3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),C3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A3))),aa(real,real,G5,C3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A3))),aa(real,real,F6,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_6589_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,G: fun(A,B),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,A3)),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_zg(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_6590_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A3),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A3)),D6) )
                     => ( aa(A,B,F2,X4) != aa(A,B,F2,A3) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xh(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_6591_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A3),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$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),A3))),D6) )
                     => ( aa(A,B,F2,X4) != aa(A,B,F2,A3) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ue(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_6592_GMVT,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),G: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( ! [X4: real] :
            ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),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))),F2) )
       => ( ! [X4: real] :
              ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X4,top_top(set(real)))) )
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),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))),G) )
           => ( ! [X4: real] :
                  ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X4,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C3,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),C3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A3))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A3))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_6593_sequentially__imp__eventually__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,A3: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( ! [F3: fun(nat,A)] :
                ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,F3,N5))
               => ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N5)),A3)
                 => ( aa(fun(nat,A),$o,order_mono(nat,A),F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_xw(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_6594_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q2: fun(A,B),C2: B,Ta: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_zh(fun(A,B),fun(B,fun(A,B)),Q2),C2),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_6595_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C2: B,Q2: fun(A,B),Ta: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zi(B,fun(fun(A,B),fun(A,B)),C2),Q2),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_6596_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F2: fun(A,B),A4: A,B4: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B4))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A4)),aa(A,B,F2,B4))) ) ) ).

% mono_inf
tff(fact_6597_mono__Suc,axiom,
    aa(fun(nat,nat),$o,order_mono(nat,nat),suc) ).

% mono_Suc
tff(fact_6598_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F4)
         => ( differentiable(A,B,G,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% differentiable_add
tff(fact_6599_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A] : aa(fun(A,A),$o,order_mono(A,A),aa(A,fun(A,A),plus_plus(A),A3)) ) ).

% mono_add
tff(fact_6600_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Nb: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => aa(fun(A,A),$o,order_mono(A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ) ) ).

% mono_pow
tff(fact_6601_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( aa(fun(A,A),$o,order_mono(A,A),Q)
         => aa(fun(nat,A),$o,order_mono(nat,A),aTP_Lamp_zj(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_6602_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% mono_strict_invE
tff(fact_6603_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( aa(fun(nat,A),$o,order_mono(nat,A),F2)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4))) ) ) ).

% incseq_Suc_iff
tff(fact_6604_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),aa(nat,A,X6,aa(nat,nat,suc,N)))
         => aa(fun(nat,A),$o,order_mono(nat,A),X6) ) ) ).

% incseq_SucI
tff(fact_6605_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: fun(nat,A),I: nat] :
          ( aa(fun(nat,A),$o,order_mono(nat,A),A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A4,I)),aa(nat,A,A4,aa(nat,nat,suc,I))) ) ) ).

% incseq_SucD
tff(fact_6606_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% monoD
tff(fact_6607_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% monoE
tff(fact_6608_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) )
         => aa(fun(A,B),$o,order_mono(A,B),F2) ) ) ).

% monoI
tff(fact_6609_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
        <=> ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) ) ) ) ).

% mono_def
tff(fact_6610_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( aa(fun(nat,A),$o,order_mono(nat,A),F2)
         => ( 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,F2,I)),aa(nat,A,F2,J)) ) ) ) ).

% incseqD
tff(fact_6611_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( aa(fun(nat,A),$o,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_6612_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mono_invE
tff(fact_6613_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),A4: A,B4: A,Nb: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),A4)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),B4)) ) ) ) ).

% funpow_mono
tff(fact_6614_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(fun(nat,nat),$o,order_mono(nat,nat),aa(nat,fun(nat,nat),times_times(nat),Nb)) ) ).

% mono_times_nat
tff(fact_6615_differentiable__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xb,S))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% differentiable_mult
tff(fact_6616_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => aa(fun(A,A),$o,order_mono(A,A),aa(A,fun(A,A),times_times(A),A3)) ) ) ).

% mono_mult
tff(fact_6617_differentiable__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A),Nb: nat] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => differentiable(A,B,aa(nat,fun(A,B),aTP_Lamp_rt(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% differentiable_power
tff(fact_6618_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F2: fun(A,B),Mb: A,Nb: A,M4: B,N3: B] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( ( image(A,B,F2,set_or7035219750837199246ssThan(A,Mb,Nb)) = set_or7035219750837199246ssThan(B,M4,N3) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb)
             => ( aa(A,B,F2,Mb) = M4 ) ) ) ) ) ).

% mono_image_least
tff(fact_6619_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F2: fun(A,A),P2: A,K: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,F2,P2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_6620_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F2: fun(A,A),P2: A,K: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,P2)),P2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A))),P2) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_6621_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),I: nat,J: nat,Xb: A,Y: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( 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,F2,Xb))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I),F2),Xb)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J),F2),Y)) ) ) ) ) ) ).

% funpow_mono2
tff(fact_6622_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A4: fun(C,A),I5: set(C)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_zk(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),image(C,A,A4,I5)))) ) ) ).

% mono_SUP
tff(fact_6623_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ).

% mono_Sup
tff(fact_6624_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A4))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))) ) ) ).

% mono_Inf
tff(fact_6625_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F2: fun(A,B),A4: fun(C,A),I5: set(C)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),image(C,A,A4,I5)))),aa(set(B),B,complete_Inf_Inf(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_zk(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4),I5))) ) ) ).

% mono_INF
tff(fact_6626_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( aa(fun(A,A),$o,order_mono(A,A),Q)
         => order_antimono(nat,A,aTP_Lamp_zl(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_6627_incseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L5: A,Nb: nat] :
          ( aa(fun(nat,A),$o,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_6628_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ro(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% differentiable_divide
tff(fact_6629_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Mb: nat,Nb: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2),top_top(A))) ) ) ) ).

% funpow_increasing
tff(fact_6630_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Mb: nat,Nb: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),bot_bot(A))) ) ) ) ).

% funpow_decreasing
tff(fact_6631_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => aa(fun(nat,nat),$o,order_mono(nat,nat),aTP_Lamp_zm(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_6632_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( finite_finite(A,image(nat,A,F2,top_top(set(nat))))
         => ( aa(fun(nat,A),$o,order_mono(nat,A),F2)
           => ( ! [N: nat] :
                  ( ( aa(nat,A,F2,N) = aa(nat,A,F2,aa(nat,nat,suc,N)) )
                 => ( aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,F2,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) )
             => ? [N8: nat] :
                  ( ! [N5: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N8)
                     => ! [M: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N8)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N5)
                           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,M)),aa(nat,A,F2,N5)) ) ) )
                  & ! [N5: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N5)
                     => ( aa(nat,A,F2,N8) = aa(nat,A,F2,N5) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_6633_tendsto__at__left__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [B2: A,A3: A,X6: fun(A,B),L5: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( ! [S4: fun(nat,A)] :
                ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S4,N5)),A3)
               => ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,S4,N5))
                 => ( aa(fun(nat,A),$o,order_mono(nat,A),S4)
                   => ( filterlim(nat,A,S4,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_xx(fun(A,B),fun(fun(nat,A),fun(nat,B)),X6),S4),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X6,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3))) ) ) ) ).

% tendsto_at_left_sequentially
tff(fact_6634_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F5: fun(nat,nat)] :
          ( aa(fun(nat,nat),$o,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)) )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
             => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),aa(nat,nat,F5,I4)) ) )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) )
              <=> ( aa(nat,nat,F5,I4) = aa(nat,nat,F5,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_6635_MVT,axiom,
    ! [A3: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
       => ( ! [X4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
               => differentiable(real,real,F2,topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ? [L3: real,Z3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z3)
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
              & has_field_derivative(real,F2,L3,topolo174197925503356063within(real,Z3,top_top(set(real))))
              & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)),L3) ) ) ) ) ) ).

% MVT
tff(fact_6636_IVT_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),A3: B,Y: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,A3)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,B2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A3,B2),F2)
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT'
tff(fact_6637_IVT2_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),B2: B,Y: A,A3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,B2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A3,B2),F2)
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_6638_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_6639_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_6640_continuous__on__mult__const,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [S: set(A),C2: A] : topolo81223032696312382ous_on(A,A,S,aa(A,fun(A,A),times_times(A),C2)) ) ).

% continuous_on_mult_const
tff(fact_6641_continuous__on__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),C2: B] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => topolo81223032696312382ous_on(A,B,S,aa(B,fun(A,B),aTP_Lamp_zn(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_on_mult_right
tff(fact_6642_continuous__on__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),C2: B] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => topolo81223032696312382ous_on(A,B,S,aa(B,fun(A,B),aTP_Lamp_zo(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_on_mult_left
tff(fact_6643_continuous__on__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A4,F2)
         => ( topolo81223032696312382ous_on(A,B,A4,G)
           => topolo81223032696312382ous_on(A,B,A4,aa(fun(A,B),fun(A,B),aTP_Lamp_zp(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_mult'
tff(fact_6644_continuous__on__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_zq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_mult
tff(fact_6645_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => ( ! [X4: A] :
                  ( member(A,X4,S)
                 => ( aa(A,B,G,X4) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_zr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_6646_continuous__on__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_zs(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_add
tff(fact_6647_continuous__on__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),Nb: nat] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => topolo81223032696312382ous_on(A,B,S,aa(nat,fun(A,B),aTP_Lamp_zt(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% continuous_on_power
tff(fact_6648_continuous__on__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo81223032696312382ous_on(A,B,A4,F2)
         => ( topolo81223032696312382ous_on(A,nat,A4,G)
           => topolo81223032696312382ous_on(A,B,A4,aa(fun(A,nat),fun(A,B),aTP_Lamp_zu(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_power'
tff(fact_6649_continuous__on__real__sqrt,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_zv(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_real_sqrt
tff(fact_6650_continuous__on__real__root,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),Nb: nat] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => topolo81223032696312382ous_on(A,real,S,aa(nat,fun(A,real),aTP_Lamp_zw(fun(A,real),fun(nat,fun(A,real)),F2),Nb)) ) ) ).

% continuous_on_real_root
tff(fact_6651_continuous__on__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,C,S,G)
           => topolo81223032696312382ous_on(A,product_prod(B,C),S,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_zx(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_on_Pair
tff(fact_6652_continuous__onI__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & dense_order(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(B,A),A4: set(B)] :
          ( topolo1002775350975398744n_open(A,image(B,A,F2,A4))
         => ( ! [X4: B,Y4: B] :
                ( member(B,X4,A4)
               => ( member(B,Y4,A4)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y4)) ) ) )
           => topolo81223032696312382ous_on(B,A,A4,F2) ) ) ) ).

% continuous_onI_mono
tff(fact_6653_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_zy(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_6654_open__Collect__less__Int,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ? [A8: set(A)] :
                ( topolo1002775350975398744n_open(A,A8)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),S) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,real),fun(A,$o),aa(fun(A,real),fun(fun(A,real),fun(A,$o)),aTP_Lamp_zz(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,$o))),S),F2),G)) ) ) ) ) ) ).

% open_Collect_less_Int
tff(fact_6655_open__Collect__positive,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ? [A8: set(A)] :
              ( topolo1002775350975398744n_open(A,A8)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),S) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,real),fun(A,$o),aTP_Lamp_aaa(set(A),fun(fun(A,real),fun(A,$o)),S),F2)) ) ) ) ) ).

% open_Collect_positive
tff(fact_6656_continuous__on__powr_H,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X4: A] :
                  ( member(A,X4,S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F2,X4))
                    & ( ( aa(A,real,F2,X4) = zero_zero(real) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X4)) ) ) )
             => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_aab(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_6657_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X4: A] :
                  ( member(A,X4,S)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,X4)) )
             => ( ! [X4: A] :
                    ( member(A,X4,S)
                   => ( aa(A,real,F2,X4) != one_one(real) ) )
               => ( ! [X4: A] :
                      ( member(A,X4,S)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X4)) )
                 => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_aac(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_6658_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_6659_remdups__adj__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ( remdups_adj(A,Xs) = aa(list(A),list(A),aa(A,fun(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_6660_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,B2: A,F2: fun(A,A)] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X4)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B2)
               => ? [Y5: A] : has_field_derivative(A,F2,Y5,topolo174197925503356063within(A,X4,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A3,B2),F2) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_6661_Rolle__deriv,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),F6: fun(real,fun(real,real))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( ( aa(real,real,F2,A3) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
         => ( ! [X4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
                 => has_derivative(real,real,F2,aa(real,fun(real,real),F6,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
           => ? [Z3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
                & ! [X5: real] : aa(real,real,aa(real,fun(real,real),F6,Z3),X5) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_6662_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => aa(fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),$o,order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),aTP_Lamp_aad(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ).

% lexordp.mono
tff(fact_6663_mvt,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),F6: fun(real,fun(real,real))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
       => ( ! [X4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
               => has_derivative(real,real,F2,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),A3),Xi)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xi),B2)
                 => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A3)) != aa(real,real,aa(real,fun(real,real),F6,Xi),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)) ) ) ) ) ) ) ).

% mvt
tff(fact_6664_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_6665_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B2),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_6666_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B2),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_6667_DERIV__isconst__end,axiom,
    ! [A3: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
       => ( ! [X4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ( aa(real,real,F2,B2) = aa(real,real,F2,A3) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_6668_DERIV__neg__imp__decreasing__open,axiom,
    ! [A3: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),zero_zero(real)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A3)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_6669_DERIV__pos__imp__increasing__open,axiom,
    ! [A3: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y5) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,A3)),aa(real,real,F2,B2)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_6670_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),hd(A,Xs)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_6671_DERIV__isconst2,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
       => ( ! [X4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Xb)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),B2)
             => ( aa(real,real,F2,Xb) = aa(real,real,F2,A3) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_6672_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A,B2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3)))
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)))
           => ( ! [X4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),B2)
                   => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X4)),topolo174197925503356063within(A,X4,top_top(set(A)))) ) )
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B2),F2) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_6673_Rolle,axiom,
    ! [A3: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B2)
     => ( ( aa(real,real,F2,A3) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
         => ( ! [X4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
                 => differentiable(real,real,F2,topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
           => ? [Z3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
                & has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_6674_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,real,F4,G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aae(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aae(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aae(A,A))))
                 => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_zd(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_6675_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_pj(nat,fun(real,real),Nb),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_6676_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_6677_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_aaf(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6678_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B4: set(A),A4: set(A)] :
      ( inj_on(A,B,F2,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),image(A,B,F2,A4)),image(A,B,F2,B4)) ) ) ).

% inj_on_strict_subset
tff(fact_6679_inj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( inj_on(A,A,F2,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(set(A))) ) ).

% inj_fn
tff(fact_6680_inj__add__left,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A3),top_top(set(A))) ) ).

% inj_add_left
tff(fact_6681_option_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),top_top(set(option(A)))) ) ).

% option.inj_map
tff(fact_6682_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A4: set(A)] : inj_on(A,A,aTP_Lamp_aag(A,fun(A,A),A3),A4) ) ).

% inj_on_add'
tff(fact_6683_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A4: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A3),A4) ) ).

% inj_on_add
tff(fact_6684_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,A4: set(A)] :
          ( ( A3 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),A4) ) ) ).

% inj_on_mult
tff(fact_6685_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
             => ( aa(A,B,F2,X4) != aa(A,B,F2,Y4) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_6686_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X4: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
             => ( member(A,X4,A4)
               => ( member(A,Y4,A4)
                 => ( aa(A,B,F2,X4) != aa(A,B,F2,Y4) ) ) ) )
         => ( ! [X4: A,Y4: A] :
                ( member(A,X4,A4)
               => ( member(A,Y4,A4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X4) ) ) )
           => inj_on(A,B,F2,A4) ) ) ) ).

% linorder_inj_onI
tff(fact_6687_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),image(B,A,F2,A4))),aa(set(B),nat,finite_card(B),A4))
     => ~ inj_on(B,A,F2,A4) ) ).

% pigeonhole
tff(fact_6688_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A3: A,Xb: A,B2: A,F2: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B2),F2)
             => ( inj_on(A,B,F2,set_or1337092689740270186AtMost(A,A3,B2))
               => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),aa(A,B,F2,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,B2)) )
                  | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),aa(A,B,F2,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,A3)) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_6689_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aae(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_zc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_6690_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B4: set(B)] :
      ( finite_finite(A,A4)
     => ( finite_finite(B,B4)
       => ( ? [F5: fun(A,B)] :
              ( inj_on(A,B,F5,A4)
              & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,F5,A4)),B4) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_6691_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B4: set(B)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,F2,A4)),B4)
       => ( finite_finite(B,B4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ).

% card_inj_on_le
tff(fact_6692_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B4: set(B)] :
      ( finite_finite(A,A4)
     => ( finite_finite(B,B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B4))
         => ? [F3: fun(A,B)] :
              ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,F3,A4)),B4)
              & inj_on(A,B,F3,A4) ) ) ) ) ).

% card_le_inj
tff(fact_6693_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_6694_funpow__inj__finite,axiom,
    ! [A: $tType,P2: fun(A,A),Xb: A] :
      ( inj_on(A,A,P2,top_top(set(A)))
     => ( finite_finite(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aah(fun(A,A),fun(A,fun(A,$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_6695_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aae(A,A))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aai(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_6696_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),Ta: list(product_prod(A,C)),K: A,Xb: C] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,Ta),K) = aa(C,option(C),some(C),Xb) )
       => ( aa(B,option(C),map_of(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_aaj(fun(A,B),fun(A,fun(C,product_prod(B,C))),F2)),Ta)),aa(A,B,F2,K)) = aa(C,option(C),some(C),Xb) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_6697_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_wm(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,F2,at_top(real),F4)
      <=> filterlim(A,real,aa(fun(A,real),fun(A,real),comp(real,real,A,inverse_inverse(real)),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_6698_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_aak(int,fun(A,A),Nb),aa(int,fun(A,A),aTP_Lamp_aal(A,fun(int,fun(A,A)),Xb),Nb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_power_int'
tff(fact_6699_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_6700_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_6701_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_6702_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_6703_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_6704_case__map__option,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: A,H: fun(B,A),F2: fun(C,B),Xb: option(C)] : aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),case_option(A,B),G),H),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F2),Xb)) = aa(option(C),A,aa(fun(C,A),fun(option(C),A),aa(A,fun(fun(C,A),fun(option(C),A)),case_option(A,C),G),aa(fun(C,B),fun(C,A),comp(B,A,C,H),F2)),Xb) ).

% case_map_option
tff(fact_6705_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_6706_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_6707_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_6708_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_6709_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_6710_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_6711_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_6712_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_6713_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$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,$o),ord_less_eq(A),power_int(A,A3,Nb)),power_int(A,B2,Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_6714_power__int__minus__left__odd,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int,A3: 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),A3),Nb) = aa(A,A,uminus_uminus(A),power_int(A,A3,Nb)) ) ) ) ).

% power_int_minus_left_odd
tff(fact_6715_power__int__minus__left__even,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int,A3: 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),A3),Nb) = power_int(A,A3,Nb) ) ) ) ).

% power_int_minus_left_even
tff(fact_6716_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_6717_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A4: 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_aam(A,fun(B,product_prod(B,A)))),A4) ).

% swap_inj_on
tff(fact_6718_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X6: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_aan(fun(A,B),fun(A,product_prod(A,B)),F2),X6) ).

% inj_on_convol_ident
tff(fact_6719_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_6720_inj__Suc,axiom,
    ! [N2: set(nat)] : inj_on(nat,nat,suc,N2) ).

% inj_Suc
tff(fact_6721_inj__Some,axiom,
    ! [A: $tType,A4: set(A)] : inj_on(A,option(A),some(A),A4) ).

% inj_Some
tff(fact_6722_inj__on__diff__nat,axiom,
    ! [N2: set(nat),K: nat] :
      ( ! [N: nat] :
          ( member(nat,N,N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) )
     => inj_on(nat,nat,aTP_Lamp_ji(nat,fun(nat,nat),K),N2) ) ).

% inj_on_diff_nat
tff(fact_6723_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_6724_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_6725_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_6726_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_6727_funpow__Suc__right,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)),F2) ).

% funpow_Suc_right
tff(fact_6728_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,F2),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ).

% funpow.simps(2)
tff(fact_6729_power__int__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Xb),Nb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),power_int(A,Xb,Nb)) ) ).

% power_int_one_over
tff(fact_6730_comp__funpow,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F2: fun(B,B)] : aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aa(nat,fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),compow(fun(fun(A,B),fun(A,B))),Nb),comp(B,B,A,F2)) = comp(B,B,A,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Nb),F2)) ).

% comp_funpow
tff(fact_6731_power__int__divide__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),Mb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,Xb,Mb)),power_int(A,Y,Mb)) ) ).

% power_int_divide_distrib
tff(fact_6732_map__option_Ocompositionality,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),Option: option(C)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),G),Option)) = aa(option(C),option(A),aa(fun(C,A),fun(option(C),option(A)),map_option(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,F2),G)),Option) ).

% map_option.compositionality
tff(fact_6733_option_Omap__comp,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(B,A),F2: fun(C,B),V: option(C)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),G),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F2),V)) = aa(option(C),option(A),aa(fun(C,A),fun(option(C),option(A)),map_option(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,G),F2)),V) ).

% option.map_comp
tff(fact_6734_map__option_Ocomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : aa(fun(option(A),option(C)),fun(option(A),option(B)),comp(option(C),option(B),option(A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F2)),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),G)) = aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),aa(fun(A,C),fun(A,B),comp(C,B,A,F2),G)) ).

% map_option.comp
tff(fact_6735_funpow__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ).

% funpow_add
tff(fact_6736_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_6737_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N2: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A3,Nb)),power_int(A,A3,N2)) ) ) ) ).

% power_int_increasing
tff(fact_6738_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A4: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X4: B,Y4: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X4)),aa(B,A,H,Y4))
           => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,H),G)),A4) = aa(B,A,H,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),A4)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_6739_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N2: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,Nb)),power_int(A,A3,N2)) ) ) ) ).

% power_int_strict_increasing
tff(fact_6740_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_6741_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Mb: int,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Mb != Nb ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,Xb,Mb)),power_int(A,Xb,Nb)) ) ) ) ).

% power_int_diff
tff(fact_6742_power__int__minus__one__diff__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: 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),A3),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),A3)) ) ).

% power_int_minus_one_diff_commute
tff(fact_6743_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite(A,A4)
     => ? [N: nat,F3: fun(nat,A)] :
          ( ( A4 = image(nat,A,F3,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),N))) )
          & inj_on(nat,A,F3,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6744_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite(A,A4)
     => ? [F3: fun(A,nat),N: nat] :
          ( ( image(A,nat,F3,A4) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),N)) )
          & inj_on(A,nat,F3,A4) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6745_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N2: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,N2)),power_int(A,A3,Nb)) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_6746_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_6747_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(int,$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,A3,Nb)) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_6748_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_6749_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(int,$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,A3,Nb)) ) ) ) ).

% one_less_power_int
tff(fact_6750_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_6751_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),H: fun(A,B),G: fun(B,C)] :
          ( finite_finite(A,A4)
         => ( ! [X4: A,Y4: A] :
                ( member(A,X4,A4)
               => ( member(A,Y4,A4)
                 => ( ( X4 != Y4 )
                   => ( ( aa(A,B,H,X4) = aa(A,B,H,Y4) )
                     => ( aa(B,C,G,aa(A,B,H,X4)) = one_one(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),image(A,B,H,A4)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),H)),A4) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_6752_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xb: A,S: set(A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),image(A,A,G,S)))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_image_chain
tff(fact_6753_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,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,F2),aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_6754_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xb: A,Db: A,S: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_chain
tff(fact_6755_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X4: nat] :
            ( member(nat,X4,I5)
           => aa(nat,$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),I5) ) ) ).

% inj_on_nth
tff(fact_6756_power__int__minus__left__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( division_ring(C)
        & one(A)
        & uminus(A) )
     => ! [Xb: B,A3: 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),A3),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,A3,Nb)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_6757_inj__on__funpow__least,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A),S: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),S) = S )
     => ( ! [M2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M2),F2),S) != S ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_aao(fun(A,A),fun(A,fun(nat,A)),F2),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% inj_on_funpow_least
tff(fact_6758_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(int,$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,A3,Nb)) ) ) ) ) ).

% power_int_antimono
tff(fact_6759_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(int,$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,A3,Nb)),power_int(A,B2,Nb)) ) ) ) ) ).

% power_int_strict_mono
tff(fact_6760_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N2: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
             => ( ( ( A3 != zero_zero(A) )
                  | ( N2 != zero_zero(int) )
                  | ( Nb = zero_zero(int) ) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A3,N2)),power_int(A,A3,Nb)) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_6761_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_6762_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),F2: fun(A,C)] :
          ( finite_finite(A,I5)
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G,aa(A,C,F2,I2))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),image(A,C,F2,I5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,G),F2)),I5)) ) ) ) ).

% sum_image_le
tff(fact_6763_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_6764_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),A3: A,D2: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D2)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D2),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_6765_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),D2: A,F4: filter(B),A3: A] :
          ( filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D2)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D2),top_top(set(A))))
        <=> filterlim(A,B,F2,F4,topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_6766_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_6767_power__int__minus__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Nb: int] :
          power_int(A,aa(A,A,uminus_uminus(A),A3),Nb) = $ite(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Nb),power_int(A,A3,Nb),aa(A,A,uminus_uminus(A),power_int(A,A3,Nb))) ) ).

% power_int_minus_left
tff(fact_6768_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_6769_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_6770_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_6771_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_6772_powr__real__of__int_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( ( Xb != zero_zero(real) )
          | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb) )
       => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),Nb)) = power_int(real,Xb,Nb) ) ) ) ).

% powr_real_of_int'
tff(fact_6773_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_6774_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xb: A,S: set(A),Nb: int] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_aap(fun(A,A),fun(int,fun(A,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Nb)),power_int(A,aa(A,A,F2,Xb),aa(int,int,aa(int,fun(int,int),minus_minus(int),Nb),one_one(int))))),D2),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_power_int
tff(fact_6775_has__derivative__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(B,A),Xb: B,F6: fun(B,A),S3: set(B),Nb: int] :
          ( ( aa(B,A,F2,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F6,topolo174197925503356063within(B,Xb,S3))
           => has_derivative(B,A,aa(int,fun(B,A),aTP_Lamp_aaq(fun(B,A),fun(int,fun(B,A)),F2),Nb),aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_aar(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),F2),Xb),F6),Nb),topolo174197925503356063within(B,Xb,S3)) ) ) ) ).

% has_derivative_power_int
tff(fact_6776_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A3) )
           => ( linorder_insort_key(A,A,aTP_Lamp_ov(A,A),A3,Xs) = append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_6777_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_6778_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% sup.bounded_iff
tff(fact_6779_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_6780_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),filter2(A,P,map(B,A,F2,Xs))) = aa(list(B),nat,size_size(list(B)),filter2(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P),F2),Xs)) ).

% length_filter_map
tff(fact_6781_length__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),linorder_insort_key(A,B,F2,Xb,Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_insort
tff(fact_6782_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,B),X5: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F2)),aTP_Lamp_aas(A,option(C))),X5) = none(B) ).

% map_option_o_empty
tff(fact_6783_option_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,nat),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),comp(option(B),nat,option(A),size_option(B,F2)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G)) = size_option(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F2),G)) ).

% option.size_gen_o_map
tff(fact_6784_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_6785_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_nh(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_6786_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_aat(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_6787_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_nh(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_6788_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_aat(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_6789_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_6790_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_6791_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% less_supI1
tff(fact_6792_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,B2: A,A3: A] :
          ( aa(A,$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),A3),B2)) ) ) ).

% less_supI2
tff(fact_6793_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb3
tff(fact_6794_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_6795_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% sup.strict_boundedE
tff(fact_6796_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_6797_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% sup.strict_coboundedI1
tff(fact_6798_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% sup.strict_coboundedI2
tff(fact_6799_insort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Y: A,Ys: list(A)] :
          linorder_insort_key(A,B,F2,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),linorder_insort_key(A,B,F2,Xb,Ys))) ) ).

% insort_key.simps(2)
tff(fact_6800_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% sup.coboundedI2
tff(fact_6801_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% sup.coboundedI1
tff(fact_6802_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_6803_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb_iff1
tff(fact_6804_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ).

% sup.cobounded2
tff(fact_6805_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ).

% sup.cobounded1
tff(fact_6806_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.order_iff
tff(fact_6807_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3) ) ) ) ).

% sup.boundedI
tff(fact_6808_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% sup.boundedE
tff(fact_6809_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_6810_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_6811_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_6812_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb1
tff(fact_6813_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X4: A,Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,aa(A,fun(A,A),F2,X4),Y4))
         => ( ! [X4: A,Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),aa(A,A,aa(A,fun(A,A),F2,X4),Y4))
           => ( ! [X4: A,Y4: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),X4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y4),Z3)),X4) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F2,Xb),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_6814_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% sup.orderI
tff(fact_6815_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.orderE
tff(fact_6816_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_6817_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_6818_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)) ) ) ) ).

% sup_mono
tff(fact_6819_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).

% sup.mono
tff(fact_6820_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,B2: A,A3: A] :
          ( aa(A,$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),A3),B2)) ) ) ).

% le_supI2
tff(fact_6821_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% le_supI1
tff(fact_6822_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_6823_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_6824_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xb)
         => ( aa(A,$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),A3),B2)),Xb) ) ) ) ).

% le_supI
tff(fact_6825_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),Xb)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xb)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb) ) ) ) ).

% le_supE
tff(fact_6826_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_6827_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_6828_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_6829_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_6830_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_6831_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F2: fun(A,B),A4: A,B4: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A4)),aa(A,B,F2,B4))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B4))) ) ) ).

% mono_sup
tff(fact_6832_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),linorder_insort_key(A,A,aTP_Lamp_ov(A,A),Xb,Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_6833_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_6834_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P2: A,Q2: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q2),R))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P2),aa(A,A,uminus_uminus(A),Q2))),R) ) ) ).

% sup_neg_inf
tff(fact_6835_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_6836_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_6837_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B4: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) ) ).

% less_eq_Inf_inter
tff(fact_6838_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_6839_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_6840_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_6841_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_6842_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A3: A] : bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A3)) = aa(fun(nat,nat),fun(nat,$o),comp(nat,$o,nat,bit_se5641148757651400278ts_bit(A,A3)),aa(nat,fun(nat,nat),plus_plus(nat),Nb)) ) ).

% bit_drop_bit_eq
tff(fact_6843_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_6844_card__Un__le,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4))) ).

% card_Un_le
tff(fact_6845_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_6846_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: 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),K)) = 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),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_6847_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_6848_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B),A3: A] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),aa(A,B,F2,X4)) )
         => ( linorder_insort_key(A,B,F2,A3,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_6849_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_6850_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_6851_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_6852_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),comp(A,A,nat,inverse_inverse(A)),F2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_aau(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_6853_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xb: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,linorder_insort_key(B,A,F2,Xb,Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs)) ) ) ).

% sorted_insort_key
tff(fact_6854_prod_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B4: set(set(A)),G: fun(A,B)] :
          ( ! [X4: set(A)] :
              ( member(set(A),X4,B4)
             => finite_finite(A,X4) )
         => ( ! [A13: set(A)] :
                ( member(set(A),A13,B4)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B4)
                   => ( ( A13 != A24 )
                     => ! [X4: A] :
                          ( member(A,X4,A13)
                         => ( member(A,X4,A24)
                           => ( aa(A,B,G,X4) = one_one(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),B4) ) ) ) ) ).

% prod.Union_comp
tff(fact_6855_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A4: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A4) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G),zero_zero(A),A4) ) ).

% sum.eq_fold
tff(fact_6856_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,times_times(A)),G),one_one(A),A4) ) ).

% prod.eq_fold
tff(fact_6857_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_6858_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B4)) ) ) ) ) ).

% sum.union_inter
tff(fact_6859_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ).

% prod.union_inter
tff(fact_6860_card__Un__Int,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( finite_finite(A,A4)
     => ( finite_finite(A,B4)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) ) ) ) ).

% card_Un_Int
tff(fact_6861_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_6862_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_6863_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_6864_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_6865_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_6866_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_6867_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_6868_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_aam(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_6869_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_nl(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_6870_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: A,B4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),aa(set(A),A,complete_Sup_Sup(A),image(nat,A,aTP_Lamp_yl(A,fun(nat,A),B4),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B4) ) ).

% SUP_nat_binary
tff(fact_6871_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_6872_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,$o),Xb: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => ( aa(B,$o,P,Xb)
           => ( filter2(B,P,linorder_insort_key(B,A,F2,Xb,Xs)) = linorder_insort_key(B,A,F2,Xb,filter2(B,P,Xs)) ) ) ) ) ).

% filter_insort
tff(fact_6873_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_aav(A,fun(A,$o)),aTP_Lamp_aaw(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_6874_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_6875_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_6876_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_6877_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_6878_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_6879_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_6880_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))
                 => ( aa(A,B,G,X4) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B4)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_6881_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A4: set(A),B4: set(A),F2: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) ) ) ) ) ).

% sum_Un
tff(fact_6882_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4) = bot_bot(set(A)) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B4)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6883_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6884_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6885_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B4: set(A),F2: fun(A,B)] :
          ( finite_finite(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4))
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A4)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) ) ) ) ).

% sum_Un2
tff(fact_6886_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A4)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) ) ) ) ) ).

% sum.union_diff2
tff(fact_6887_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_6888_card__Un__disjoint,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( finite_finite(A,A4)
     => ( finite_finite(A,B4)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4) = bot_bot(set(A)) )
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6889_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) ) ) ) ) ).

% prod.union_diff2
tff(fact_6890_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_6891_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_6892_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_6893_sum__Un__nat,axiom,
    ! [A: $tType,A4: set(A),B4: set(A),F2: fun(A,nat)] :
      ( finite_finite(A,A4)
     => ( finite_finite(A,B4)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A4)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),B4))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_6894_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_6895_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_6896_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_6897_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_6898_map__of__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),Xs: list(product_prod(A,C))] : map_of(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_mz(fun(C,B),fun(A,fun(C,product_prod(A,B))),F2)),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F2)),map_of(A,C,Xs)) ).

% map_of_map
tff(fact_6899_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xs: list(B)] : map_filter(B,A,F2,Xs) = map(B,A,aa(fun(B,option(A)),fun(B,A),comp(option(A),A,B,the2(A)),F2),filter2(B,aTP_Lamp_aax(fun(B,option(A)),fun(B,$o),F2),Xs)) ).

% map_filter_def
tff(fact_6900_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A4: set(A),B4: set(A),F2: fun(A,B)] :
          ( finite_finite(A,A4)
         => ( finite_finite(A,B4)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))
                 => ( aa(A,B,F2,X4) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) ) ) ) ) ) ).

% prod_Un
tff(fact_6901_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A),Ys: list(A)] :
          ( inj_on(A,B,F2,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),map(A,B,F2,Xs))
           => ( distinct(B,map(A,B,F2,Xs))
             => ( sorted_wrt(B,ord_less_eq(B),map(A,B,F2,Ys))
               => ( distinct(B,map(A,B,F2,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_6902_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = aa(fun(num,int),fun(num,int),comp(int,int,num,uminus_uminus(int)),numeral_numeral(int)) ).

% Code_Target_Int.negative_def
tff(fact_6903_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A3: A,Xs: list(A),F2: fun(A,B)] :
          ( member(A,A3,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),map(A,B,F2,Xs))
           => ( ( hd(A,filter2(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aay(A,fun(fun(A,B),fun(A,$o)),A3),F2),Xs)) = A3 )
             => ( linorder_insort_key(A,B,F2,A3,remove1(A,A3,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_6904_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B)),X5: A,Xa: B] :
      ( aa(B,$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_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R2),S3)) ) ).

% sup_Un_eq2
tff(fact_6905_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_6906_sup__enat__def,axiom,
    sup_sup(extended_enat) = ord_max(extended_enat) ).

% sup_enat_def
tff(fact_6907_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A3,Xs)) ) ) ).

% sorted_remove1
tff(fact_6908_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Xb: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F2,remove1(B,Xb,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_6909_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(B,C))] :
      ( finite_finite(product_prod(A,B),R2)
     => ( finite_finite(product_prod(B,C),S3)
       => ( relcomp(A,B,C,R2,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_aba(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))),R2) ) ) ) ).

% relcomp_fold
tff(fact_6910_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_6911_sum__list__map__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xb: A,Xs: list(A),F2: fun(A,B)] :
          ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ( aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F2,Xs)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xb)),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F2,remove1(A,Xb,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_6912_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B)),Xb: product_prod(C,A),R2: set(product_prod(C,A))] :
      ( 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),R2),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_abb(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xb)),relcomp(C,A,B,R2,S3),S3) ) ) ).

% insert_relcomp_fold
tff(fact_6913_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,Xs: list(A)] :
          ( member(A,A3,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( linorder_insort_key(A,A,aTP_Lamp_ov(A,A),A3,remove1(A,A3,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_6914_Id__on__fold,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite(A,A4)
     => ( id_on(A,A4) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_abc(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A4) ) ) ).

% Id_on_fold
tff(fact_6915_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))] :
      ( 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_abb(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_6916_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B4: set(A),Xb: B,A4: set(product_prod(B,A))] :
      ( finite_finite(A,B4)
     => ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),image(A,set(product_prod(B,A)),aTP_Lamp_abd(B,fun(A,set(product_prod(B,A))),Xb),B4))),A4) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_abe(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Xb),A4,B4) ) ) ).

% fold_union_pair
tff(fact_6917_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Ks: list(A)] : map_of(A,B,map(A,product_prod(A,B),aTP_Lamp_aan(fun(A,B),fun(A,product_prod(A,B)),F2),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F2),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_6918_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A)] : lattic5882676163264333800up_fin(A,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_abf(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Sup_fin.eq_fold'
tff(fact_6919_restrict__out,axiom,
    ! [A: $tType,B: $tType,Xb: A,A4: set(A),Mb: fun(A,option(B))] :
      ( ~ member(A,Xb,A4)
     => ( aa(A,option(B),restrict_map(A,B,Mb,A4),Xb) = none(B) ) ) ).

% restrict_out
tff(fact_6920_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D5: set(A),X5: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_on(A,option(B)),D5),X5) = none(B) ).

% restrict_map_empty
tff(fact_6921_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),X5: A] : aa(A,option(B),restrict_map(A,B,Mb,bot_bot(set(A))),X5) = none(B) ).

% restrict_map_to_empty
tff(fact_6922_restrict__map__def,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),A4: set(A),X5: A] :
      aa(A,option(B),restrict_map(A,B,Mb,A4),X5) = $ite(member(A,X5,A4),aa(A,option(B),Mb,X5),none(B)) ).

% restrict_map_def
tff(fact_6923_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite(A,A4)
         => ( member(A,A3,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),lattic5882676163264333800up_fin(A,A4)) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_6924_ran__restrictD,axiom,
    ! [B: $tType,A: $tType,Y: A,Mb: fun(B,option(A)),A4: set(B)] :
      ( member(A,Y,ran(B,A,restrict_map(B,A,Mb,A4)))
     => ? [X4: B] :
          ( member(B,X4,A4)
          & ( aa(B,option(A),Mb,X4) = aa(A,option(A),some(A),Y) ) ) ) ).

% ran_restrictD
tff(fact_6925_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A4)),Xb)
             => ! [A9: A] :
                  ( member(A,A9,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),Xb) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_6926_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),Xb) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A4)),Xb) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_6927_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A4)),Xb)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xb) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_6928_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A)] :
          ( ~ finite_finite(A,A4)
         => ( lattic5882676163264333800up_fin(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_6929_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( finite_finite(A,B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A4)),lattic5882676163264333800up_fin(A,B4)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_6930_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A)] :
          ( finite_finite(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A4)),lattic5882676163264333800up_fin(A,A4)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_6931_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B))] :
      ( 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_abh(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_6932_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),D5: set(A),Mb: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D5)
       => ( restrict_map(A,B,map_upds(A,B,Mb,Xs,Ys),D5) = 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)),D5),aa(list(A),set(A),set2(A),Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6933_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,append(A,Xs,Zs),Ys) = map_upds(A,B,Mb,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_6934_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,append(B,Ys,Zs)) = map_upds(A,B,Mb,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_6935_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_6936_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B4: set(A)] :
      ( finite_finite(A,B4)
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_abi(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B4)) ) ).

% comp_fun_commute_product_fold
tff(fact_6937_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,append(A,Xs,aa(list(A),list(A),aa(A,fun(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_6938_relpow__finite__bounded1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K: nat] :
      ( finite_finite(product_prod(A,A),R2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abj(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abk(set(product_prod(A,A)),fun(nat,$o),R2))))) ) ) ).

% relpow_finite_bounded1
tff(fact_6939_relpow__1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),one_one(nat)),R2) = R2 ).

% relpow_1
tff(fact_6940_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,Xb: A,X5: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_on(A,option(B)),Xb,none(B)),X5) = none(B) ).

% empty_upd_none
tff(fact_6941_image__map__upd,axiom,
    ! [B: $tType,A: $tType,Xb: A,A4: set(A),Mb: fun(A,option(B)),Y: B] :
      ( ~ member(A,Xb,A4)
     => ( image(A,option(B),fun_upd(A,option(B),Mb,Xb,aa(B,option(B),some(B),Y)),A4) = image(A,option(B),Mb,A4) ) ) ).

% image_map_upd
tff(fact_6942_finite__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Nb: nat] :
      ( finite_finite(product_prod(A,A),R2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => 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),R2)) ) ) ).

% finite_relpow
tff(fact_6943_map__upds__Cons,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),A3: A,As: list(A),B2: B,Bs: list(B)] : map_upds(A,B,Mb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs)) = map_upds(A,B,fun_upd(A,option(B),Mb,A3,aa(B,option(B),some(B),B2)),As,Bs) ).

% map_upds_Cons
tff(fact_6944_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A3: A,As: list(A),Mb: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ member(A,A3,aa(list(A),set(A),set2(A),As))
     => ( map_upds(A,B,fun_upd(A,option(B),Mb,A3,aa(B,option(B),some(B),B2)),As,Bs) = fun_upd(A,option(B),map_upds(A,B,Mb,As,Bs),A3,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_6945_map__option__o__map__upd,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,B),Mb: fun(A,option(C)),A3: A,B2: C] : aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F2)),fun_upd(A,option(C),Mb,A3,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,aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F2)),Mb),A3,aa(B,option(B),some(B),aa(C,B,F2,B2))) ).

% map_option_o_map_upd
tff(fact_6946_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),Mb,A3) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),Mb,A3,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),insert(A,B2),ran(B,A,Mb)) ) ) ).

% ran_map_upd
tff(fact_6947_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),D5: set(A),Xb: A] :
      fun_upd(A,option(B),restrict_map(A,B,Mb,D5),Xb,none(B)) = $ite(member(A,Xb,D5),restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),restrict_map(A,B,Mb,D5)) ).

% fun_upd_None_restrict
tff(fact_6948_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_6949_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,Ta: fun(A,option(B)),K: A,Xb: B] :
      ~ ! [X4: A] : aa(A,option(B),fun_upd(A,option(B),Ta,K,aa(B,option(B),some(B),Xb)),X4) = none(B) ).

% map_upd_nonempty
tff(fact_6950_map__upd__Some__unfold,axiom,
    ! [B: $tType,A: $tType,Mb: fun(B,option(A)),A3: B,B2: A,Xb: B,Y: A] :
      ( ( aa(B,option(A),fun_upd(B,option(A),Mb,A3,aa(A,option(A),some(A),B2)),Xb) = aa(A,option(A),some(A),Y) )
    <=> ( ( ( Xb = A3 )
          & ( B2 = Y ) )
        | ( ( Xb != A3 )
          & ( aa(B,option(A),Mb,Xb) = aa(A,option(A),some(A),Y) ) ) ) ) ).

% map_upd_Some_unfold
tff(fact_6951_map__upd__triv,axiom,
    ! [A: $tType,B: $tType,Ta: fun(B,option(A)),K: B,Xb: A] :
      ( ( aa(B,option(A),Ta,K) = aa(A,option(A),some(A),Xb) )
     => ( fun_upd(B,option(A),Ta,K,aa(A,option(A),some(A),Xb)) = Ta ) ) ).

% map_upd_triv
tff(fact_6952_map__upd__eqD1,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),A3: A,Xb: B,Nb: fun(A,option(B)),Y: B] :
      ( ( fun_upd(A,option(B),Mb,A3,aa(B,option(B),some(B),Xb)) = fun_upd(A,option(B),Nb,A3,aa(B,option(B),some(B),Y)) )
     => ( Xb = Y ) ) ).

% map_upd_eqD1
tff(fact_6953_relpow__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),R2) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Mb),R2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ).

% relpow_add
tff(fact_6954_relpow__Suc__I2,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: 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),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),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2)) ) ) ).

% relpow_Suc_I2
tff(fact_6955_relpow__Suc__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,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),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2))
     => ~ ! [Y4: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4),R2)
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ) ) ).

% relpow_Suc_E2
tff(fact_6956_relpow__Suc__D2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,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),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2))
     => ? [Y4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4),R2)
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ) ) ).

% relpow_Suc_D2
tff(fact_6957_relpow__Suc__I,axiom,
    ! [A: $tType,Xb: A,Y: A,Nb: nat,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),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z),R2)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2)) ) ) ).

% relpow_Suc_I
tff(fact_6958_relpow__Suc__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,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),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2))
     => ~ ! [Y4: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z),R2) ) ) ).

% relpow_Suc_E
tff(fact_6959_relpow__0__E,axiom,
    ! [A: $tType,Xb: A,Y: 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),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2))
     => ( Xb = Y ) ) ).

% relpow_0_E
tff(fact_6960_relpow__0__I,axiom,
    ! [A: $tType,Xb: 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),Xb),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2)) ).

% relpow_0_I
tff(fact_6961_relpow__Suc__D2_H,axiom,
    ! [A: $tType,Nb: nat,R2: set(product_prod(A,A)),X5: A,Y5: 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),X5),Y5),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z4),R2) )
     => ? [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),X5),W2),R2)
          & 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),R2)) ) ) ).

% relpow_Suc_D2'
tff(fact_6962_relpowp__relpow__eq,axiom,
    ! [A: $tType,Nb: nat,R2: set(product_prod(A,A)),X5: A,Xa: A] :
      ( aa(A,$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_abl(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X5),Xa)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ) ).

% relpowp_relpow_eq
tff(fact_6963_relpow__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,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),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y4: A,M2: nat] :
              ( ( Nb = aa(nat,nat,suc,M2) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4),R2)
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M2),R2)) ) ) ) ) ).

% relpow_E2
tff(fact_6964_relpow__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,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),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y4: A,M2: nat] :
              ( ( Nb = aa(nat,nat,suc,M2) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M2),R2))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z),R2) ) ) ) ) ).

% relpow_E
tff(fact_6965_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_6966_relpow__fun__conv,axiom,
    ! [A: $tType,A3: A,B2: A,Nb: nat,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = A3 )
          & ( aa(nat,A,F5,Nb) = B2 )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),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,I4)),aa(nat,A,F5,aa(nat,nat,suc,I4))),R2) ) ) ) ).

% relpow_fun_conv
tff(fact_6967_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),A3: B,B2: A] :
      ( finite_finite(option(A),image(B,option(A),F2,top_top(set(B))))
     => finite_finite(option(A),image(B,option(A),fun_upd(B,option(A),F2,A3,aa(A,option(A),some(A),B2)),top_top(set(B)))) ) ).

% finite_range_updI
tff(fact_6968_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_6969_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xb: A] : restrict_map(A,B,F2,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),F2,Xb,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_6970_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      map_upds(A,B,fun_upd(A,option(B),F2,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,F2,Xs,Ys),fun_upd(A,option(B),map_upds(A,B,F2,Xs,Ys),Xb,aa(B,option(B),some(B),Y))) ).

% map_upd_upds_conv_if
tff(fact_6971_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)),aa(product_prod(A,B),fun(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_6972_relpow__finite__bounded,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K: nat] :
      ( finite_finite(product_prod(A,A),R2)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abj(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abm(set(product_prod(A,A)),fun(nat,$o),R2))))) ) ).

% relpow_finite_bounded
tff(fact_6973_ntrancl__def,axiom,
    ! [A: $tType,Nb: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,Nb,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abj(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abn(nat,fun(nat,$o),Nb)))) ).

% ntrancl_def
tff(fact_6974_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( finite_finite(product_prod(A,A),R2)
     => ( transitive_trancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abj(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abk(set(product_prod(A,A)),fun(nat,$o),R2)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_6975_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_trancl(product_prod(A,B),R))
     => ( ! [A5: A,B3: 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),A5),B3)),R)
           => aa(B,$o,aa(A,fun(B,$o),P,A5),B3) )
       => ( ! [A5: A,B3: 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),A5),B3)),transitive_trancl(product_prod(A,B),R))
             => ( 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),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R)
               => ( aa(B,$o,aa(A,fun(B,$o),P,A5),B3)
                 => aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_6976_converse__trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R))
     => ( ! [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),B2),R)
           => aa(A,$o,P,Y4) )
       => ( ! [Y4: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2),transitive_trancl(A,R))
               => ( aa(A,$o,P,Z3)
                 => aa(A,$o,P,Y4) ) ) )
         => aa(A,$o,P,A3) ) ) ) ).

% converse_trancl_induct
tff(fact_6977_trancl__trans__induct,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),P: fun(A,fun(A,$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,R))
     => ( ! [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),X4),Y4),R)
           => aa(A,$o,aa(A,fun(A,$o),P,X4),Y4) )
       => ( ! [X4: A,Y4: 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),Y4),transitive_trancl(A,R))
             => ( aa(A,$o,aa(A,fun(A,$o),P,X4),Y4)
               => ( 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),transitive_trancl(A,R))
                 => ( aa(A,$o,aa(A,fun(A,$o),P,Y4),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_6978_trancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),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),A3),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),C2),transitive_trancl(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),A3),C2),transitive_trancl(A,R)) ) ) ).

% trancl_into_trancl2
tff(fact_6979_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),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),A3),B2),transitive_trancl(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),B2),C2),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_6980_irrefl__trancl__rD,axiom,
    ! [A: $tType,R: 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,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),R)
       => ( Xb != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_6981_converse__tranclE,axiom,
    ! [A: $tType,Xb: A,Z: 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),Z),transitive_trancl(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),Xb),Z),R)
       => ~ ! [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),Xb),Y4),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),Z),transitive_trancl(A,R)) ) ) ) ).

% converse_tranclE
tff(fact_6982_r__r__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),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),A3),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),C2),R2)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R2)) ) ) ).

% r_r_into_trancl
tff(fact_6983_trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R))
     => ( ! [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),A3),Y4),R)
           => aa(A,$o,P,Y4) )
       => ( ! [Y4: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y4),transitive_trancl(A,R))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R)
               => ( aa(A,$o,P,Y4)
                 => aa(A,$o,P,Z3) ) ) )
         => aa(A,$o,P,B2) ) ) ) ).

% trancl_induct
tff(fact_6984_trancl__trans,axiom,
    ! [A: $tType,Xb: A,Y: A,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),transitive_trancl(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),Y),Z),transitive_trancl(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),Xb),Z),transitive_trancl(A,R)) ) ) ).

% trancl_trans
tff(fact_6985_tranclE,axiom,
    ! [A: $tType,A3: 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),A3),B2),transitive_trancl(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),A3),B2),R)
       => ~ ! [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),A3),C3),transitive_trancl(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),C3),B2),R) ) ) ) ).

% tranclE
tff(fact_6986_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A3: 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),A3),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),A3),B2),transitive_trancl(A,R)) ) ).

% trancl.r_into_trancl
tff(fact_6987_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: 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),A1),A22),transitive_trancl(A,R))
    <=> ( ? [A6: A,B5: A] :
            ( ( A1 = A6 )
            & ( A22 = B5 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5),R) )
        | ? [A6: A,B5: 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),B5),transitive_trancl(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),B5),C4),R) ) ) ) ).

% trancl.simps
tff(fact_6988_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: 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),A1),A22),transitive_trancl(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),A1),A22),R)
       => ~ ! [B3: 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),B3),transitive_trancl(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),B3),A22),R) ) ) ) ).

% trancl.cases
tff(fact_6989_finite__trancl__ntranl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( finite_finite(product_prod(A,A),R2)
     => ( transitive_trancl(A,R2) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),R2)),one_one(nat)),R2) ) ) ).

% finite_trancl_ntranl
tff(fact_6990_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_6991_trancl__power,axiom,
    ! [A: $tType,P2: product_prod(A,A),R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),P2,transitive_trancl(A,R2))
    <=> ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
          & member(product_prod(A,A),P2,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N4),R2)) ) ) ).

% trancl_power
tff(fact_6992_trancl__insert2,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abo(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A3),B2),R)))) ).

% trancl_insert2
tff(fact_6993_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_6994_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),K: A,V: B] : graph(A,B,fun_upd(A,option(B),Mb,K,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),K),V)),graph(A,B,fun_upd(A,option(B),Mb,K,none(B)))) ).

% graph_map_upd
tff(fact_6995_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( ( dom(A,B,F2) = bot_bot(set(A)) )
    <=> ! [X: A] : aa(A,option(B),F2,X) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_6996_fun__upd__None__if__notin__dom,axiom,
    ! [B: $tType,A: $tType,K: A,Mb: fun(A,option(B))] :
      ( ~ member(A,K,dom(A,B,Mb))
     => ( fun_upd(A,option(B),Mb,K,none(B)) = Mb ) ) ).

% fun_upd_None_if_notin_dom
tff(fact_6997_dom__const,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : dom(A,B,aTP_Lamp_abp(fun(A,B),fun(A,option(B)),F2)) = top_top(set(A)) ).

% dom_const
tff(fact_6998_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_on(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_6999_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_on(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_7000_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_7001_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),Xb: A,Y: option(B)] :
      dom(A,B,fun_upd(A,option(B),F2,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,F2)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))),aa(set(A),set(A),insert(A,Xb),dom(A,B,F2))) ).

% dom_fun_upd
tff(fact_7002_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_7003_in__graphI,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),K: B,V: A] :
      ( ( aa(B,option(A),Mb,K) = 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),K),V),graph(B,A,Mb)) ) ).

% in_graphI
tff(fact_7004_in__graphD,axiom,
    ! [A: $tType,B: $tType,K: 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),K),V),graph(A,B,Mb))
     => ( aa(A,option(B),Mb,K) = aa(B,option(B),some(B),V) ) ) ).

% in_graphD
tff(fact_7005_insert__dom,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xb: B,Y: A] :
      ( ( aa(B,option(A),F2,Xb) = aa(A,option(A),some(A),Y) )
     => ( aa(set(B),set(B),insert(B,Xb),dom(B,A,F2)) = dom(B,A,F2) ) ) ).

% insert_dom
tff(fact_7006_domI,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),Mb,A3) = aa(A,option(A),some(A),B2) )
     => member(B,A3,dom(B,A,Mb)) ) ).

% domI
tff(fact_7007_domD,axiom,
    ! [A: $tType,B: $tType,A3: A,Mb: fun(A,option(B))] :
      ( member(A,A3,dom(A,B,Mb))
     => ? [B3: B] : aa(A,option(B),Mb,A3) = aa(B,option(B),some(B),B3) ) ).

% domD
tff(fact_7008_domIff,axiom,
    ! [A: $tType,B: $tType,A3: A,Mb: fun(A,option(B))] :
      ( member(A,A3,dom(A,B,Mb))
    <=> ( aa(A,option(B),Mb,A3) != none(B) ) ) ).

% domIff
tff(fact_7009_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_abq(fun(A,option(B)),fun(A,$o),Mb)) ).

% dom_def
tff(fact_7010_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_abr(fun(A,option(B)),fun(A,product_prod(A,B)),Mb),dom(A,B,Mb)) ).

% graph_eq_to_snd_dom
tff(fact_7011_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K: A,V: B,Mb: fun(A,option(B)),A4: 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),K),V),graph(A,B,restrict_map(A,B,Mb,A4)))
     => member(A,K,A4) ) ).

% graph_restrictD(1)
tff(fact_7012_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( finite_finite(A,dom(A,B,F2))
     => ( ~ finite_finite(A,top_top(set(A)))
       => ? [X4: A] : aa(A,option(B),F2,X4) = none(B) ) ) ).

% finite_map_freshness
tff(fact_7013_dom__minus,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xb: B,A4: set(B)] :
      ( ( aa(B,option(A),F2,Xb) = none(A) )
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F2)),aa(set(B),set(B),insert(B,Xb),A4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F2)),A4) ) ) ).

% dom_minus
tff(fact_7014_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K: A,V: B,Mb: fun(A,option(B)),A4: 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),K),V),graph(A,B,restrict_map(A,B,Mb,A4)))
     => ( aa(A,option(B),Mb,K) = aa(B,option(B),some(B),V) ) ) ).

% graph_restrictD(2)
tff(fact_7015_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_abs(fun(A,option(B)),fun(product_prod(A,B),$o),Mb)) ).

% graph_def
tff(fact_7016_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),P: fun(fun(A,option(B)),$o)] :
      ( finite_finite(A,dom(A,B,Mb))
     => ( aa(fun(A,option(B)),$o,P,aTP_Lamp_on(A,option(B)))
       => ( ! [K3: A,V3: B,M2: fun(A,option(B))] :
              ( finite_finite(A,dom(A,B,M2))
             => ( ~ member(A,K3,dom(A,B,M2))
               => ( aa(fun(A,option(B)),$o,P,M2)
                 => aa(fun(A,option(B)),$o,P,fun_upd(A,option(B),M2,K3,aa(B,option(B),some(B),V3))) ) ) )
         => aa(fun(A,option(B)),$o,P,Mb) ) ) ) ).

% finite_Map_induct
tff(fact_7017_graph__fun__upd__None,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),K: A] : graph(A,B,fun_upd(A,option(B),Mb,K,none(B))) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(A,fun(product_prod(A,B),$o),aTP_Lamp_abt(fun(A,option(B)),fun(A,fun(product_prod(A,B),$o)),Mb),K)) ).

% graph_fun_upd_None
tff(fact_7018_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xb: A] :
      ( ( dom(A,B,F2) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) )
    <=> ? [V5: B] : F2 = fun_upd(A,option(B),aTP_Lamp_on(A,option(B)),Xb,aa(B,option(B),some(B),V5)) ) ).

% dom_eq_singleton_conv
tff(fact_7019_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,map(A,product_prod(A,B),aTP_Lamp_abr(fun(A,option(B)),fun(A,product_prod(A,B)),Mb),Xs)) = Mb ) ) ).

% map_of_map_keys
tff(fact_7020_option_Orec__o__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: B,Ga: fun(C,B),F2: fun(A,C)] : aa(fun(option(A),option(C)),fun(option(A),B),comp(option(C),B,option(A),aa(fun(C,B),fun(option(C),B),aa(B,fun(fun(C,B),fun(option(C),B)),rec_option(B,C),G),Ga)),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),F2)) = aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),rec_option(B,A),G),aa(fun(A,C),fun(A,B),aTP_Lamp_abu(fun(C,B),fun(fun(A,C),fun(A,B)),Ga),F2)) ).

% option.rec_o_map
tff(fact_7021_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_abw(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_7022_option_Osimps_I7_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),X22: B] : aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),rec_option(A,B),F1),F22),aa(B,option(B),some(B),X22)) = aa(B,A,F22,X22) ).

% option.simps(7)
tff(fact_7023_option_Osimps_I6_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,A)] : aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),rec_option(A,B),F1),F22),none(B)) = F1 ).

% option.simps(6)
tff(fact_7024_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( finite_finite(product_prod(A,A),R2)
     => ( transitive_rtrancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abj(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abm(set(product_prod(A,A)),fun(nat,$o),R2)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_7025_listrel__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : listrel(A,B,X5) = aa(fun(product_prod(list(A),list(B)),$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_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),X5)))) ).

% listrel_def
tff(fact_7026_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay)
       => ( ! [A5: A,B3: 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),A5),B3)),transitive_rtrancl(product_prod(A,B),R))
             => ( 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),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R)
               => ( aa(B,$o,aa(A,fun(B,$o),P,A5),B3)
                 => aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_7027_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xaa: A,Xb: B,Za: A,Zb: B,R: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xaa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)),transitive_rtrancl(product_prod(A,B),R))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xaa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
       => ~ ! [A5: A,B3: 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),A5),B3)),R)
             => ~ 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),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)),transitive_rtrancl(product_prod(A,B),R)) ) ) ) ).

% converse_rtranclE2
tff(fact_7028_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Bx),By)
       => ( ! [A5: A,B3: 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),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R)
             => ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R))
               => ( aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba)
                 => aa(B,$o,aa(A,fun(B,$o),P,A5),B3) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_7029_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: 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),A1),A22),transitive_rtrancl(A,R))
     => ( ( A22 != A1 )
       => ~ ! [B3: 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),B3),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),B3),A22),R) ) ) ) ).

% rtrancl.cases
tff(fact_7030_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: 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),A1),A22),transitive_rtrancl(A,R))
    <=> ( ? [A6: A] :
            ( ( A1 = A6 )
            & ( A22 = A6 ) )
        | ? [A6: A,B5: 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),B5),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),B5),C4),R) ) ) ) ).

% rtrancl.simps
tff(fact_7031_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A3: 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),A3),A3),transitive_rtrancl(A,R)) ).

% rtrancl.rtrancl_refl
tff(fact_7032_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),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),A3),B2),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),B2),C2),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_rtrancl(A,R)) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_7033_rtranclE,axiom,
    ! [A: $tType,A3: 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),A3),B2),transitive_rtrancl(A,R))
     => ( ( A3 != B2 )
       => ~ ! [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),A3),Y4),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),Y4),B2),R) ) ) ) ).

% rtranclE
tff(fact_7034_rtrancl__trans,axiom,
    ! [A: $tType,Xb: A,Y: A,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),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),Y),Z),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),Xb),Z),transitive_rtrancl(A,R)) ) ) ).

% rtrancl_trans
tff(fact_7035_rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R))
     => ( aa(A,$o,P,A3)
       => ( ! [Y4: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y4),transitive_rtrancl(A,R))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R)
               => ( aa(A,$o,P,Y4)
                 => aa(A,$o,P,Z3) ) ) )
         => aa(A,$o,P,B2) ) ) ) ).

% rtrancl_induct
tff(fact_7036_converse__rtranclE,axiom,
    ! [A: $tType,Xb: A,Z: 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),Z),transitive_rtrancl(A,R))
     => ( ( Xb != Z )
       => ~ ! [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),Xb),Y4),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),Z),transitive_rtrancl(A,R)) ) ) ) ).

% converse_rtranclE
tff(fact_7037_converse__rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R))
     => ( aa(A,$o,P,B2)
       => ( ! [Y4: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2),transitive_rtrancl(A,R))
               => ( aa(A,$o,P,Z3)
                 => aa(A,$o,P,Y4) ) ) )
         => aa(A,$o,P,A3) ) ) ) ).

% converse_rtrancl_induct
tff(fact_7038_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),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),A3),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),C2),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),A3),C2),transitive_rtrancl(A,R)) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_7039_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,Xb: A,Y: A,R: 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,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),transitive_rtrancl(list(A),listrel1(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(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),transitive_rtrancl(list(A),listrel1(A,R))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_7040_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A3: 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),A3),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,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),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),Y4),X4),Q)
             => ( Y4 = X4 ) ) )
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,P)) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_7041_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A3: 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),A3),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,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),A3),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),Y4),Q)
             => ( X4 = Y4 ) ) )
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,P)) ) ) ).

% rtrancl_Un_separatorE
tff(fact_7042_tranclD,axiom,
    ! [A: $tType,Xb: A,Y: 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),Y),transitive_trancl(A,R2))
     => ? [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),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),Y),transitive_rtrancl(A,R2)) ) ) ).

% tranclD
tff(fact_7043_rtranclD,axiom,
    ! [A: $tType,A3: 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),A3),B2),transitive_rtrancl(A,R2))
     => ( ( A3 = B2 )
        | ( ( A3 != B2 )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2)) ) ) ) ).

% rtranclD
tff(fact_7044_tranclD2,axiom,
    ! [A: $tType,Xb: A,Y: 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),Y),transitive_trancl(A,R2))
     => ? [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,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),Y),R2) ) ) ).

% tranclD2
tff(fact_7045_trancl__into__rtrancl,axiom,
    ! [A: $tType,A3: 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),A3),B2),transitive_trancl(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),A3),B2),transitive_rtrancl(A,R)) ) ).

% trancl_into_rtrancl
tff(fact_7046_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,Xb: A,Y: 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),Y),transitive_rtrancl(A,R2))
    <=> ( ( 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,R2)) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_7047_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),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),A3),B2),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),B2),C2),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R)) ) ) ).

% rtrancl_into_trancl1
tff(fact_7048_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),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),A3),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),C2),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),A3),C2),transitive_trancl(A,R)) ) ) ).

% rtrancl_into_trancl2
tff(fact_7049_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,Xb: A,Y: A,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),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),Y),Z),transitive_trancl(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),Xb),Z),transitive_trancl(A,R)) ) ) ).

% rtrancl_trancl_trancl
tff(fact_7050_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),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),A3),B2),transitive_trancl(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),B2),C2),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),A3),C2),transitive_trancl(A,R)) ) ) ).

% trancl_rtrancl_trancl
tff(fact_7051_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R: 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,R)))
     => ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_7052_rtrancl__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abx(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A3),B2),R)))) ).

% rtrancl_insert
tff(fact_7053_trancl__insert,axiom,
    ! [A: $tType,Y: A,Xb: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xb)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abx(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Y),Xb),R)))) ).

% trancl_insert
tff(fact_7054_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),X5: list(A),Xa: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),X5),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)),X5),Xa),listrel(A,B,R)) ) ).

% listrelp_listrel_eq
tff(fact_7055_pos__deriv__imp__strict__mono,axiom,
    ! [F2: fun(real,real),F6: fun(real,real)] :
      ( ! [X4: real] : has_field_derivative(real,F2,aa(real,real,F6,X4),topolo174197925503356063within(real,X4,top_top(set(real))))
     => ( ! [X4: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F6,X4))
       => order_strict_mono(real,real,F2) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_7056_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] :
      size_list(A,F2,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,F2,hd(A,Xs))),size_list(A,F2,tl(A,Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_7057_length__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(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_7058_tl__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: 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_7059_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% strict_mono_less_eq
tff(fact_7060_strict__mono__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [R: fun(A,B),Mb: A,Nb: A] :
          ( order_strict_mono(A,B,R)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),Nb)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,R,Mb)),aa(A,B,R,Nb)) ) ) ) ).

% strict_mono_leD
tff(fact_7061_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),tl(A,Xs)) ) ) ).

% sorted_tl
tff(fact_7062_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% strict_monoD
tff(fact_7063_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4)) )
         => order_strict_mono(A,B,F2) ) ) ).

% strict_monoI
tff(fact_7064_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
        <=> ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) ) ) ) ).

% strict_mono_def
tff(fact_7065_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% strict_mono_less
tff(fact_7066_strict__mono__add,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A] : order_strict_mono(A,A,aTP_Lamp_in(A,fun(A,A),K)) ) ).

% strict_mono_add
tff(fact_7067_tl__take,axiom,
    ! [A: $tType,Nb: nat,Xs: 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)),tl(A,Xs)) ).

% tl_take
tff(fact_7068_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)),tl(A,Xs)))) ).

% Nitpick.size_list_simp(2)
tff(fact_7069_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)),tl(A,Xs)))
     => ( aa(nat,A,nth(A,tl(A,Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,Nb)) ) ) ).

% nth_tl
tff(fact_7070_list__encode_Opelims,axiom,
    ! [Xb: list(nat),Y: 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),aa(nat,fun(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),nat_list_encode(Xs2)))) )
                 => ~ accp(list(nat),nat_list_encode_rel,aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X4),Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_7071_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_7072_strict__mono__imp__increasing,axiom,
    ! [F2: fun(nat,nat),Nb: nat] :
      ( order_strict_mono(nat,nat,F2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,F2,Nb)) ) ).

% strict_mono_imp_increasing
tff(fact_7073_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_strict_mono(nat,A,F2)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4))) ) ) ).

% strict_mono_Suc_iff
tff(fact_7074_total__on__def,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( total_on(A,A4,R)
    <=> ! [X: A] :
          ( member(A,X,A4)
         => ! [Xa2: A] :
              ( member(A,Xa2,A4)
             => ( ( X != Xa2 )
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa2),R)
                  | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X),R) ) ) ) ) ) ).

% total_on_def
tff(fact_7075_total__onI,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( ! [X4: A,Y4: A] :
          ( member(A,X4,A4)
         => ( member(A,Y4,A4)
           => ( ( X4 != Y4 )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4),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),X4),R) ) ) ) )
     => total_on(A,A4,R) ) ).

% total_onI
tff(fact_7076_increasing__Bseq__subseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,nat)] :
          ( ! [X4: nat,Y4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Y4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,X4))),real_V7770717601297561774m_norm(A,aa(nat,A,F2,Y4))) )
         => ( order_strict_mono(nat,nat,G)
           => ( bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_aby(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F2),G),at_top(nat))
            <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ) ).

% increasing_Bseq_subseq_iff
tff(fact_7077_inverse__real_Oabs__eq,axiom,
    ! [Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
     => ( aa(real,real,inverse_inverse(real),aa(fun(nat,rat),real,real2,Xb)) = aa(fun(nat,rat),real,real2,
            $ite(vanishes(Xb),aTP_Lamp_pa(nat,rat),aTP_Lamp_ox(fun(nat,rat),fun(nat,rat),Xb))) ) ) ).

% inverse_real.abs_eq
tff(fact_7078_Field__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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),A3),B2)),R)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),field2(A,R)) ).

% Field_insert
tff(fact_7079_real_Oabs__induct,axiom,
    ! [P: fun(real,$o),Xb: real] :
      ( ! [Y4: fun(nat,rat)] :
          ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Y4),Y4)
         => aa(real,$o,P,aa(fun(nat,rat),real,real2,Y4)) )
     => aa(real,$o,P,Xb) ) ).

% real.abs_induct
tff(fact_7080_zero__real_Orsp,axiom,
    aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,aTP_Lamp_pa(nat,rat)),aTP_Lamp_pa(nat,rat)) ).

% zero_real.rsp
tff(fact_7081_one__real_Orsp,axiom,
    aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,aTP_Lamp_pb(nat,rat)),aTP_Lamp_pb(nat,rat)) ).

% one_real.rsp
tff(fact_7082_realrel__refl,axiom,
    ! [X6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X6),X6) ) ).

% realrel_refl
tff(fact_7083_FieldI2,axiom,
    ! [A: $tType,I: A,J: 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),I),J),R2)
     => member(A,J,field2(A,R2)) ) ).

% FieldI2
tff(fact_7084_FieldI1,axiom,
    ! [A: $tType,I: A,J: 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),I),J),R2)
     => member(A,I,field2(A,R2)) ) ).

% FieldI1
tff(fact_7085_uminus__real_Oabs__eq,axiom,
    ! [Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
     => ( aa(real,real,uminus_uminus(real),aa(fun(nat,rat),real,real2,Xb)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_or(fun(nat,rat),fun(nat,rat)),Xb)) ) ) ).

% uminus_real.abs_eq
tff(fact_7086_plus__real_Oabs__eq,axiom,
    ! [Xaa: fun(nat,rat),Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xaa),Xaa)
     => ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(fun(nat,rat),real,real2,Xaa)),aa(fun(nat,rat),real,real2,Xb)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ou(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Xaa),Xb)) ) ) ) ).

% plus_real.abs_eq
tff(fact_7087_times__real_Oabs__eq,axiom,
    ! [Xaa: fun(nat,rat),Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xaa),Xaa)
     => ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
       => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(fun(nat,rat),real,real2,Xaa)),aa(fun(nat,rat),real,real2,Xb)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_os(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Xaa),Xb)) ) ) ) ).

% times_real.abs_eq
tff(fact_7088_realrelI,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( aa(fun(nat,rat),$o,cauchy,Y6)
       => ( vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6))
         => aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X6),Y6) ) ) ) ).

% realrelI
tff(fact_7089_realrel__def,axiom,
    ! [X5: fun(nat,rat),Xa: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X5),Xa)
    <=> ( aa(fun(nat,rat),$o,cauchy,X5)
        & aa(fun(nat,rat),$o,cauchy,Xa)
        & vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X5),Xa)) ) ) ).

% realrel_def
tff(fact_7090_Real_Opositive_Oabs__eq,axiom,
    ! [Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
     => ( aa(real,$o,positive2,aa(fun(nat,rat),real,real2,Xb))
      <=> ? [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
            & ? [K2: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N4)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,Xb,N4)) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_7091_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_abz(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),Nb)) ).

% Field_natLeq_on
tff(fact_7092_cr__real__def,axiom,
    ! [X5: fun(nat,rat),Xa: real] :
      ( aa(real,$o,aa(fun(nat,rat),fun(real,$o),cr_real,X5),Xa)
    <=> ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X5),X5)
        & ( aa(fun(nat,rat),real,real2,X5) = Xa ) ) ) ).

% cr_real_def
tff(fact_7093_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R: set(product_prod(A,A)),As2: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R,As2)
        <=> ! [I4: A,J3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I4),J3),R)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,As2,I4)),aa(A,B,As2,J3)) ) ) ) ).

% relChain_def
tff(fact_7094_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat))) ).

% natLess_def
tff(fact_7095_cofinal__def,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( bNF_Ca7293521722713021262ofinal(A,A4,R)
    <=> ! [X: A] :
          ( member(A,X,field2(A,R))
         => ? [Xa2: A] :
              ( member(A,Xa2,A4)
              & ( X != Xa2 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa2),R) ) ) ) ).

% cofinal_def
tff(fact_7096_Under__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : order_Under(A,R,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aca(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R),A4)) ).

% Under_def
tff(fact_7097_UnderS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : order_UnderS(A,R,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_acb(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R),A4)) ).

% UnderS_def
tff(fact_7098_Above__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : order_Above(A,R,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_acc(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R),A4)) ).

% Above_def
tff(fact_7099_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_7100_Total__subset__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R),R)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),id2(A))
       => ( ( R = bot_bot(set(product_prod(A,A))) )
          | ? [A5: A] : R = 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),A5),A5)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_7101_IdI,axiom,
    ! [A: $tType,A3: A] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),id2(A)) ).

% IdI
tff(fact_7102_pair__in__Id__conv,axiom,
    ! [A: $tType,A3: 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),A3),B2),id2(A))
    <=> ( A3 = B2 ) ) ).

% pair_in_Id_conv
tff(fact_7103_IdD,axiom,
    ! [A: $tType,A3: 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),A3),B2),id2(A))
     => ( A3 = B2 ) ) ).

% IdD
tff(fact_7104_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_7105_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_acd(product_prod(A,A),$o)) ).

% Id_def
tff(fact_7106_Linear__order__in__diff__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( member(A,A3,field2(A,R))
       => ( member(A,B2,field2(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),A3),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),A3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A))) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_7107_reflcl__set__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X5: A,Xa: A] :
      ( aa(A,$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,$o))),aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),fequal(A)),X5),Xa)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),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))),R),id2(A))) ) ).

% reflcl_set_eq
tff(fact_7108_Real_Opositive_Orsp,axiom,
    aa(fun(fun(nat,rat),$o),$o,aa(fun(fun(nat,rat),$o),fun(fun(fun(nat,rat),$o),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),$o,$o,realrel,fequal($o)),aTP_Lamp_ace(fun(nat,rat),$o)),aTP_Lamp_ace(fun(nat,rat),$o)) ).

% Real.positive.rsp
tff(fact_7109_max__ext_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),R2: set(product_prod(A,A))] :
      ( finite_finite(A,X6)
     => ( finite_finite(A,Y6)
       => ( ( Y6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => ? [Xa: A] :
                    ( member(A,Xa,Y6)
                    & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa),R2) ) )
           => member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X6),Y6),max_ext(A,R2)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_7110_power__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & power(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),times_times(A)),times_times(B))
           => aa(fun(B,fun(nat,B)),$o,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),$o),bNF_rel_fun(A,B,fun(nat,A),fun(nat,B),R2,bNF_rel_fun(nat,nat,A,B,fequal(nat),R2)),power_power(A)),power_power(B)) ) ) ) ).

% power_transfer
tff(fact_7111_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B))
             => ( aa(fun(B,B),$o,aa(fun(A,A),fun(fun(B,B),$o),bNF_rel_fun(A,B,A,B,R2,R2),uminus_uminus(A)),uminus_uminus(B))
               => aa(fun(int,B),$o,aa(fun(int,A),fun(fun(int,B),$o),bNF_rel_fun(int,int,A,B,fequal(int),R2),ring_1_of_int(A)),ring_1_of_int(B)) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_7112_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B))
             => aa(fun(num,B),$o,aa(fun(num,A),fun(fun(num,B),$o),bNF_rel_fun(num,num,A,B,fequal(num),R2),numeral_numeral(A)),numeral_numeral(B)) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_7113_max__ext_Ocases,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R2))
     => ~ ( finite_finite(A,A1)
         => ( finite_finite(A,A22)
           => ( ( A22 != bot_bot(set(A)) )
             => ~ ! [X5: A] :
                    ( member(A,X5,A1)
                   => ? [Xa3: A] :
                        ( member(A,Xa3,A22)
                        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3),R2) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_7114_max__ext_Osimps,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R2))
    <=> ( finite_finite(A,A1)
        & finite_finite(A,A22)
        & ( A22 != bot_bot(set(A)) )
        & ! [X: A] :
            ( member(A,X,A1)
           => ? [Xa2: A] :
                ( member(A,Xa2,A22)
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa2),R2) ) ) ) ) ).

% max_ext.simps
tff(fact_7115_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B))
             => aa(fun(nat,B),$o,aa(fun(nat,A),fun(fun(nat,B),$o),bNF_rel_fun(nat,nat,A,B,fequal(nat),R2),semiring_1_of_nat(A)),semiring_1_of_nat(B)) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_7116_max__ext__def,axiom,
    ! [A: $tType,X5: set(product_prod(A,A))] : max_ext(A,X5) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),max_extp(A,aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,$o)),X5)))) ).

% max_ext_def
tff(fact_7117_less__eq__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less_eq(nat)),ord_less_eq(nat)) ).

% less_eq_natural.rsp
tff(fact_7118_less__integer_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),fequal(int),bNF_rel_fun(int,int,$o,$o,fequal(int),fequal($o))),ord_less(int)),ord_less(int)) ).

% less_integer.rsp
tff(fact_7119_less__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less(nat)),ord_less(nat)) ).

% less_natural.rsp
tff(fact_7120_times__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),times_times(nat)),times_times(nat)) ).

% times_natural.rsp
tff(fact_7121_num_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,$o))] : aa(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),$o,aa(fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))),fun(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),$o),bNF_rel_fun(A,B,fun(fun(num,A),fun(fun(num,A),fun(num,A))),fun(fun(num,B),fun(fun(num,B),fun(num,B))),S3,bNF_rel_fun(fun(num,A),fun(num,B),fun(fun(num,A),fun(num,A)),fun(fun(num,B),fun(num,B)),bNF_rel_fun(num,num,A,B,fequal(num),S3),bNF_rel_fun(fun(num,A),fun(num,B),fun(num,A),fun(num,B),bNF_rel_fun(num,num,A,B,fequal(num),S3),bNF_rel_fun(num,num,A,B,fequal(num),S3)))),case_num(A)),case_num(B)) ).

% num.case_transfer
tff(fact_7122_sub_Orsp,axiom,
    aa(fun(num,fun(num,int)),$o,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,int)),$o),bNF_rel_fun(num,num,fun(num,int),fun(num,int),fequal(num),bNF_rel_fun(num,num,int,int,fequal(num),fequal(int))),aTP_Lamp_acf(num,fun(num,int))),aTP_Lamp_acf(num,fun(num,int))) ).

% sub.rsp
tff(fact_7123_divide__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),divide_divide(int)),divide_divide(int)) ).

% divide_integer.rsp
tff(fact_7124_divide__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),divide_divide(nat)),divide_divide(nat)) ).

% divide_natural.rsp
tff(fact_7125_plus__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),plus_plus(nat)),plus_plus(nat)) ).

% plus_natural.rsp
tff(fact_7126_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
           => aa(fun($o,B),$o,aa(fun($o,A),fun(fun($o,B),$o),bNF_rel_fun($o,$o,A,B,fequal($o),R2),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B)) ) ) ) ).

% transfer_rule_of_bool
tff(fact_7127_plus__real_Orsp,axiom,
    aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),fun(fun(nat,rat),fun(nat,rat)),realrel,bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel)),aTP_Lamp_ou(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),aTP_Lamp_ou(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% plus_real.rsp
tff(fact_7128_uminus__real_Orsp,axiom,
    aa(fun(fun(nat,rat),fun(nat,rat)),$o,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(fun(nat,rat),fun(nat,rat)),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel),aTP_Lamp_or(fun(nat,rat),fun(nat,rat))),aTP_Lamp_or(fun(nat,rat),fun(nat,rat))) ).

% uminus_real.rsp
tff(fact_7129_times__real_Orsp,axiom,
    aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),fun(fun(nat,rat),fun(nat,rat)),realrel,bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel)),aTP_Lamp_os(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),aTP_Lamp_os(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% times_real.rsp
tff(fact_7130_inverse__real_Orsp,axiom,
    aa(fun(fun(nat,rat),fun(nat,rat)),$o,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(fun(nat,rat),fun(nat,rat)),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel),aTP_Lamp_xy(fun(nat,rat),fun(nat,rat))),aTP_Lamp_xy(fun(nat,rat),fun(nat,rat))) ).

% inverse_real.rsp
tff(fact_7131_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X5: set(A),Xa: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X5),Xa)
    <=> member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X5),Xa),max_ext(A,R2)) ) ).

% max_extp_max_ext_eq
tff(fact_7132_Real_Opositive_Otransfer,axiom,
    aa(fun(real,$o),$o,aa(fun(fun(nat,rat),$o),fun(fun(real,$o),$o),bNF_rel_fun(fun(nat,rat),real,$o,$o,pcr_real,fequal($o)),aTP_Lamp_ace(fun(nat,rat),$o)),positive2) ).

% Real.positive.transfer
tff(fact_7133_Rat_Opositive_Otransfer,axiom,
    aa(fun(rat,$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(rat,$o),$o),bNF_rel_fun(product_prod(int,int),rat,$o,$o,pcr_rat,fequal($o)),aTP_Lamp_acg(product_prod(int,int),$o)),positive) ).

% Rat.positive.transfer
tff(fact_7134_real_Orel__eq__transfer,axiom,
    aa(fun(real,fun(real,$o)),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),$o)),fun(fun(real,fun(real,$o)),$o),bNF_rel_fun(fun(nat,rat),real,fun(fun(nat,rat),$o),fun(real,$o),pcr_real,bNF_rel_fun(fun(nat,rat),real,$o,$o,pcr_real,fequal($o))),realrel),fequal(real)) ).

% real.rel_eq_transfer
tff(fact_7135_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
tff(fact_7136_zero__real_Otransfer,axiom,
    aa(real,$o,aa(fun(nat,rat),fun(real,$o),pcr_real,aTP_Lamp_pa(nat,rat)),zero_zero(real)) ).

% zero_real.transfer
tff(fact_7137_one__real_Otransfer,axiom,
    aa(real,$o,aa(fun(nat,rat),fun(real,$o),pcr_real,aTP_Lamp_pb(nat,rat)),one_one(real)) ).

% one_real.transfer
tff(fact_7138_cr__real__eq,axiom,
    ! [X5: fun(nat,rat),Xa: real] :
      ( aa(real,$o,aa(fun(nat,rat),fun(real,$o),pcr_real,X5),Xa)
    <=> ( aa(fun(nat,rat),$o,cauchy,X5)
        & ( aa(fun(nat,rat),real,real2,X5) = Xa ) ) ) ).

% cr_real_eq
tff(fact_7139_uminus__real_Otransfer,axiom,
    aa(fun(real,real),$o,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(real,real),$o),bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real),aTP_Lamp_or(fun(nat,rat),fun(nat,rat))),uminus_uminus(real)) ).

% uminus_real.transfer
tff(fact_7140_plus__real_Otransfer,axiom,
    aa(fun(real,fun(real,real)),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(real,fun(real,real)),$o),bNF_rel_fun(fun(nat,rat),real,fun(fun(nat,rat),fun(nat,rat)),fun(real,real),pcr_real,bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real)),aTP_Lamp_ou(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),plus_plus(real)) ).

% plus_real.transfer
tff(fact_7141_times__real_Otransfer,axiom,
    aa(fun(real,fun(real,real)),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(real,fun(real,real)),$o),bNF_rel_fun(fun(nat,rat),real,fun(fun(nat,rat),fun(nat,rat)),fun(real,real),pcr_real,bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real)),aTP_Lamp_os(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),times_times(real)) ).

% times_real.transfer
tff(fact_7142_inverse__real_Otransfer,axiom,
    aa(fun(real,real),$o,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(real,real),$o),bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real),aTP_Lamp_xy(fun(nat,rat),fun(nat,rat))),inverse_inverse(real)) ).

% inverse_real.transfer
tff(fact_7143_times__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),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_ld(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int)) ).

% times_int.transfer
tff(fact_7144_num_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,$o))] : aa(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),$o,aa(fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)))),fun(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),$o),bNF_rel_fun(A,B,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),S3,bNF_rel_fun(fun(num,fun(A,A)),fun(num,fun(B,B)),fun(fun(num,fun(A,A)),fun(num,A)),fun(fun(num,fun(B,B)),fun(num,B)),bNF_rel_fun(num,num,fun(A,A),fun(B,B),fequal(num),bNF_rel_fun(A,B,A,B,S3,S3)),bNF_rel_fun(fun(num,fun(A,A)),fun(num,fun(B,B)),fun(num,A),fun(num,B),bNF_rel_fun(num,num,fun(A,A),fun(B,B),fequal(num),bNF_rel_fun(A,B,A,B,S3,S3)),bNF_rel_fun(num,num,A,B,fequal(num),S3)))),rec_num(A)),rec_num(B)) ).

% num.rec_transfer
tff(fact_7145_verit__eq__simplify_I21_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X32: num] : aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),aa(num,num,bit1,X32)) = aa(A,A,aa(num,fun(A,A),F32,X32),aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),X32)) ).

% verit_eq_simplify(21)
tff(fact_7146_verit__eq__simplify_I20_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X22: num] : aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),aa(num,num,bit0,X22)) = aa(A,A,aa(num,fun(A,A),F22,X22),aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),X22)) ).

% verit_eq_simplify(20)
tff(fact_7147_verit__eq__simplify_I19_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A))] : aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),one2) = F1 ).

% verit_eq_simplify(19)
tff(fact_7148_zero__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),pcr_int,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_zero(int)) ).

% zero_int.transfer
tff(fact_7149_int__transfer,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_ach(nat,product_prod(nat,nat))),semiring_1_of_nat(int)) ).

% int_transfer
tff(fact_7150_uminus__int_Otransfer,axiom,
    aa(fun(int,int),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),$o),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),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_le(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int)) ).

% uminus_int.transfer
tff(fact_7151_one__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),pcr_int,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_one(int)) ).

% one_int.transfer
tff(fact_7152_less__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($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),$o)),aTP_Lamp_lg(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less(int)) ).

% less_int.transfer
tff(fact_7153_less__eq__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($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),$o)),aTP_Lamp_li(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less_eq(int)) ).

% less_eq_int.transfer
tff(fact_7154_plus__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),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_lk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int)) ).

% plus_int.transfer
tff(fact_7155_minus__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),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_lm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int)) ).

% minus_int.transfer
tff(fact_7156_of__rat_Otransfer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(fun(rat,A),$o,aa(fun(product_prod(int,int),A),fun(fun(rat,A),$o),bNF_rel_fun(product_prod(int,int),rat,A,A,pcr_rat,fequal(A)),aTP_Lamp_aci(product_prod(int,int),A)),field_char_0_of_rat(A)) ) ).

% of_rat.transfer
tff(fact_7157_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))
      <=> ! [A11: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),field2(A,R))
           => ( ( A11 != bot_bot(set(A)) )
             => ? [X: A] :
                  ( member(A,X,A11)
                  & ! [Xa2: A] :
                      ( member(A,Xa2,A11)
                     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa2),R) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_7158_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_7159_of__rat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( aa(rat,A,field_char_0_of_rat(A),A3) = one_one(A) )
        <=> ( A3 = one_one(rat) ) ) ) ).

% of_rat_eq_1_iff
tff(fact_7160_one__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( one_one(A) = aa(rat,A,field_char_0_of_rat(A),A3) )
        <=> ( one_one(rat) = A3 ) ) ) ).

% one_eq_of_rat_iff
tff(fact_7161_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_7162_wf__insert,axiom,
    ! [A: $tType,Y: A,Xb: A,R: 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)),R))
    <=> ( wf(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),Xb),Y),transitive_rtrancl(A,R)) ) ) ).

% wf_insert
tff(fact_7163_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R) ) ) ).

% zero_less_of_rat_iff
tff(fact_7164_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R)),zero_zero(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R),zero_zero(rat)) ) ) ).

% of_rat_less_0_iff
tff(fact_7165_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),R) ) ) ).

% one_less_of_rat_iff
tff(fact_7166_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R),one_one(rat)) ) ) ).

% of_rat_less_1_iff
tff(fact_7167_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),zero_zero(rat)),R) ) ) ).

% zero_le_of_rat_iff
tff(fact_7168_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R)),zero_zero(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R),zero_zero(rat)) ) ) ).

% of_rat_le_0_iff
tff(fact_7169_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_7170_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R),one_one(rat)) ) ) ).

% of_rat_le_1_iff
tff(fact_7171_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),R) ) ) ).

% one_le_of_rat_iff
tff(fact_7172_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_7173_wf__bounded__measure,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Ub: fun(A,nat),F2: fun(A,nat)] :
      ( ! [A5: A,B3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A5),R)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Ub,B3)),aa(A,nat,Ub,A5))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,B3)),aa(A,nat,Ub,A5))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,A5)),aa(A,nat,F2,B3)) ) )
     => wf(A,R) ) ).

% wf_bounded_measure
tff(fact_7174_of__rat__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat,S: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R)),aa(rat,A,field_char_0_of_rat(A),S))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R),S) ) ) ).

% of_rat_less_eq
tff(fact_7175_of__rat__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat,S: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R)),aa(rat,A,field_char_0_of_rat(A),S))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R),S) ) ) ).

% of_rat_less
tff(fact_7176_of__rat__dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => ? [Q3: rat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(rat,real,field_char_0_of_rat(real),Q3))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(rat,real,field_char_0_of_rat(real),Q3)),Y) ) ) ).

% of_rat_dense
tff(fact_7177_of__rat__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(rat,A,field_char_0_of_rat(A),A3)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_mult
tff(fact_7178_of__rat__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(rat,A,field_char_0_of_rat(A),A3)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_add
tff(fact_7179_of__rat__Real,axiom,
    ! [Xb: rat] : aa(rat,real,field_char_0_of_rat(real),Xb) = aa(fun(nat,rat),real,real2,aTP_Lamp_ot(rat,fun(nat,rat),Xb)) ).

% of_rat_Real
tff(fact_7180_of__rat__power,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,Nb: nat] : aa(rat,A,field_char_0_of_rat(A),aa(nat,rat,aa(rat,fun(nat,rat),power_power(rat),A3),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(rat,A,field_char_0_of_rat(A),A3)),Nb) ) ).

% of_rat_power
tff(fact_7181_of__rat__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(rat,A,field_char_0_of_rat(A),A3)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_divide
tff(fact_7182_wf__induct__rule,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
      ( wf(A,R)
     => ( ! [X4: 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),X4),R)
               => aa(A,$o,P,Y5) )
           => aa(A,$o,P,X4) )
       => aa(A,$o,P,A3) ) ) ).

% wf_induct_rule
tff(fact_7183_wf__eq__minimal,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ! [Q7: set(A)] :
          ( ? [X: A] : member(A,X,Q7)
         => ? [X: A] :
              ( member(A,X,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),X),R)
                 => ~ member(A,Y3,Q7) ) ) ) ) ).

% wf_eq_minimal
tff(fact_7184_wf__not__refl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( wf(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),A3),A3),R) ) ).

% wf_not_refl
tff(fact_7185_wf__not__sym,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,Xb: A] :
      ( wf(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),A3),Xb),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),A3),R) ) ) ).

% wf_not_sym
tff(fact_7186_wf__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( wf(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),A3),A3),R) ) ).

% wf_irrefl
tff(fact_7187_wf__induct,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
      ( wf(A,R)
     => ( ! [X4: 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),X4),R)
               => aa(A,$o,P,Y5) )
           => aa(A,$o,P,X4) )
       => aa(A,$o,P,A3) ) ) ).

% wf_induct
tff(fact_7188_wf__asym,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,Xb: A] :
      ( wf(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),A3),Xb),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),A3),R) ) ) ).

% wf_asym
tff(fact_7189_wfUNIVI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [P6: fun(A,$o),X4: A] :
          ( ! [Xa: 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),Xa),R)
                 => aa(A,$o,P6,Y4) )
             => aa(A,$o,P6,Xa) )
         => aa(A,$o,P6,X4) )
     => wf(A,R) ) ).

% wfUNIVI
tff(fact_7190_wfI__min,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X4: A,Q8: set(A)] :
          ( member(A,X4,Q8)
         => ? [Xa: A] :
              ( member(A,Xa,Q8)
              & ! [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),Xa),R2)
                 => ~ member(A,Y4,Q8) ) ) )
     => wf(A,R2) ) ).

% wfI_min
tff(fact_7191_wfE__min,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: A,Q: set(A)] :
      ( wf(A,R2)
     => ( member(A,Xb,Q)
       => ~ ! [Z3: A] :
              ( member(A,Z3,Q)
             => ~ ! [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),Z3),R2)
                   => ~ member(A,Y5,Q) ) ) ) ) ).

% wfE_min
tff(fact_7192_wf__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ! [P5: fun(A,$o)] :
          ( ! [X: 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),X),R)
                 => aa(A,$o,P5,Y3) )
             => aa(A,$o,P5,X) )
         => ! [X_1: A] : aa(A,$o,P5,X_1) ) ) ).

% wf_def
tff(fact_7193_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),F2: fun(nat,A)] :
      ( wf(A,R)
     => ~ ! [K3: 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,F2,aa(nat,nat,suc,K3))),aa(nat,A,F2,K3)),R) ) ).

% wf_no_infinite_down_chainE
tff(fact_7194_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ~ ? [F5: fun(nat,A)] :
          ! [I4: 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,F5,aa(nat,nat,suc,I4))),aa(nat,A,F5,I4)),R) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_7195_wfE__min_H,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Q: set(A)] :
      ( wf(A,R2)
     => ( ( Q != bot_bot(set(A)) )
       => ~ ! [Z3: A] :
              ( member(A,Z3,Q)
             => ~ ! [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),Z3),R2)
                   => ~ member(A,Y5,Q) ) ) ) ) ).

% wfE_min'
tff(fact_7196_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_7197_wf__if__measure,axiom,
    ! [A: $tType,P: fun(A,$o),F2: fun(A,nat),G: fun(A,A)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,aa(A,A,G,X4))),aa(A,nat,F2,X4)) )
     => wf(A,aa(fun(product_prod(A,A),$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_acj(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),P),G)))) ) ).

% wf_if_measure
tff(fact_7198_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_7199_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),P: fun(B,$o),K: B,Mb: fun(B,A)] :
      ( wf(A,R)
     => ( ! [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),X4),Y4),transitive_trancl(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),Y4),X4),transitive_rtrancl(A,R)) )
       => ( aa(B,$o,P,K)
         => ? [X4: B] :
              ( aa(B,$o,P,X4)
              & ! [Y5: B] :
                  ( aa(B,$o,P,Y5)
                 => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Mb,X4)),aa(B,A,Mb,Y5)),transitive_rtrancl(A,R)) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_7200_less__RealD,axiom,
    ! [Y6: fun(nat,rat),Xb: real] :
      ( aa(fun(nat,rat),$o,cauchy,Y6)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(fun(nat,rat),real,real2,Y6))
       => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(rat,real,field_char_0_of_rat(real),aa(nat,rat,Y6,N))) ) ) ).

% less_RealD
tff(fact_7201_Real__leI,axiom,
    ! [X6: fun(nat,rat),Y: real] :
      ( aa(fun(nat,rat),$o,cauchy,X6)
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(rat,real,field_char_0_of_rat(real),aa(nat,rat,X6,N))),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(fun(nat,rat),real,real2,X6)),Y) ) ) ).

% Real_leI
tff(fact_7202_le__RealI,axiom,
    ! [Y6: fun(nat,rat),Xb: real] :
      ( aa(fun(nat,rat),$o,cauchy,Y6)
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(rat,real,field_char_0_of_rat(real),aa(nat,rat,Y6,N)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(fun(nat,rat),real,real2,Y6)) ) ) ).

% le_RealI
tff(fact_7203_nonzero__of__rat__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [B2: rat,A3: rat] :
          ( ( B2 != zero_zero(rat) )
         => ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(rat,A,field_char_0_of_rat(A),A3)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ) ) ).

% nonzero_of_rat_divide
tff(fact_7204_wf__eq__minimal2,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ! [A11: set(A)] :
          ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),field2(A,R))
            & ( A11 != bot_bot(set(A)) ) )
         => ? [X: A] :
              ( member(A,X,A11)
              & ! [Xa2: A] :
                  ( member(A,Xa2,A11)
                 => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X),R) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_7205_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),Ub: fun(A,set(B)),F2: fun(A,set(B))] :
      ( ! [A5: A,B3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A5),R)
         => ( finite_finite(B,aa(A,set(B),Ub,A5))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Ub,B3)),aa(A,set(B),Ub,A5))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,B3)),aa(A,set(B),Ub,A5))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(A,set(B),F2,A5)),aa(A,set(B),F2,B3)) ) )
     => wf(A,R) ) ).

% wf_bounded_set
tff(fact_7206_of__rat__rat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [B2: int,A3: int] :
          ( ( B2 != zero_zero(int) )
         => ( aa(rat,A,field_char_0_of_rat(A),fract(A3,B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B2)) ) ) ) ).

% of_rat_rat
tff(fact_7207_finite__subset__wf,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite(A,A4)
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_ack(set(A),fun(set(A),fun(set(A),$o)),A4)))) ) ).

% finite_subset_wf
tff(fact_7208_of__rat_Orep__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xb: rat] : aa(rat,A,field_char_0_of_rat(A),Xb) = aa(A,A,aa(A,fun(A,A),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_7209_dependent__wf__choice,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P: fun(fun(A,B),fun(A,fun(B,$o)))] :
      ( wf(A,R2)
     => ( ! [F3: fun(A,B),G3: fun(A,B),X4: A,R3: B] :
            ( ! [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),Z4),X4),R2)
               => ( aa(A,B,F3,Z4) = aa(A,B,G3,Z4) ) )
           => ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F3),X4),R3)
            <=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,G3),X4),R3) ) )
       => ( ! [X4: A,F3: fun(A,B)] :
              ( ! [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),X4),R2)
                 => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F3),Y5),aa(A,B,F3,Y5)) )
             => ? [X_13: B] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F3),X4),X_13) )
         => ? [F3: fun(A,B)] :
            ! [X5: A] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F3),X5),aa(A,B,F3,X5)) ) ) ) ).

% dependent_wf_choice
tff(fact_7210_bsqr__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),$o)),fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),product_case_prod(product_prod(A,A),product_prod(A,A),$o),aa(fun(A,fun(A,fun(product_prod(A,A),$o))),fun(product_prod(A,A),fun(product_prod(A,A),$o)),product_case_prod(A,A,fun(product_prod(A,A),$o)),aTP_Lamp_acm(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),R)))) ).

% bsqr_def
tff(fact_7211_dependent__wellorder__choice,axiom,
    ! [A: $tType,B: $tType] :
      ( wellorder(B)
     => ! [P: fun(fun(B,A),fun(B,fun(A,$o)))] :
          ( ! [R3: A,F3: fun(B,A),G3: fun(B,A),X4: B] :
              ( ! [Y5: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y5),X4)
                 => ( aa(B,A,F3,Y5) = aa(B,A,G3,Y5) ) )
             => ( aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,F3),X4),R3)
              <=> aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,G3),X4),R3) ) )
         => ( ! [X4: B,F3: fun(B,A)] :
                ( ! [Y5: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y5),X4)
                   => aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,F3),Y5),aa(B,A,F3,Y5)) )
               => ? [X_13: A] : aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,F3),X4),X_13) )
           => ? [F3: fun(B,A)] :
              ! [X5: B] : aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,F3),X5),aa(B,A,F3,X5)) ) ) ) ).

% dependent_wellorder_choice
tff(fact_7212_times__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),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_ld(nat,fun(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_ld(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int.rsp
tff(fact_7213_max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : max_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_acn(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R2))) ).

% max_ext_eq
tff(fact_7214_intrel__iff,axiom,
    ! [Xb: nat,Y: nat,U: nat,V: nat] :
      ( aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),U),V))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),V) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).

% intrel_iff
tff(fact_7215_zero__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),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.rsp
tff(fact_7216_uminus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),product_prod(nat,nat)),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),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_le(nat,fun(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_le(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int.rsp
tff(fact_7217_one__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),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.rsp
tff(fact_7218_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A4: set(nat),B4: set(nat)] : nths(A,nths(A,Xs,A4),B4) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_acp(set(nat),fun(set(nat),fun(nat,$o)),A4),B4))) ).

% nths_nths
tff(fact_7219_intrel__def,axiom,
    intrel = 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),$o)),aTP_Lamp_acr(nat,fun(nat,fun(product_prod(nat,nat),$o)))) ).

% intrel_def
tff(fact_7220_less__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($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),$o)),aTP_Lamp_lg(nat,fun(nat,fun(product_prod(nat,nat),$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),$o)),aTP_Lamp_lg(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int.rsp
tff(fact_7221_less__eq__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($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),$o)),aTP_Lamp_li(nat,fun(nat,fun(product_prod(nat,nat),$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),$o)),aTP_Lamp_li(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int.rsp
tff(fact_7222_minus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),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_lm(nat,fun(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_lm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int.rsp
tff(fact_7223_plus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),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_lk(nat,fun(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_lk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int.rsp
tff(fact_7224_map__project__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),A4: set(B)] : map_project(B,A,F2,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_acs(fun(B,option(A)),fun(set(B),fun(A,$o)),F2),A4)) ).

% map_project_def
tff(fact_7225_min__ext__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : min_ext(A,R) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_act(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),R)) ).

% min_ext_def
tff(fact_7226_mono__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType] :
      ( ( order(B)
        & order(D)
        & order(C)
        & order(A) )
     => ! [A4: fun(A,fun(B,$o)),B4: fun(C,fun(D,$o))] :
          ( bi_total(A,B,A4)
         => ( aa(fun(B,fun(B,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(B,fun(B,$o)),$o),bNF_rel_fun(A,B,fun(A,$o),fun(B,$o),A4,bNF_rel_fun(A,B,$o,$o,A4,fequal($o))),ord_less_eq(A)),ord_less_eq(B))
           => ( aa(fun(D,fun(D,$o)),$o,aa(fun(C,fun(C,$o)),fun(fun(D,fun(D,$o)),$o),bNF_rel_fun(C,D,fun(C,$o),fun(D,$o),B4,bNF_rel_fun(C,D,$o,$o,B4,fequal($o))),ord_less_eq(C)),ord_less_eq(D))
             => aa(fun(fun(B,D),$o),$o,aa(fun(fun(A,C),$o),fun(fun(fun(B,D),$o),$o),bNF_rel_fun(fun(A,C),fun(B,D),$o,$o,bNF_rel_fun(A,B,C,D,A4,B4),fequal($o)),order_mono(A,C)),order_mono(B,D)) ) ) ) ) ).

% mono_transfer
tff(fact_7227_power_Opower__eq__if,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),P2: A,Mb: nat] :
      aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),P2),Mb) = $ite(Mb = zero_zero(nat),One,aa(A,A,aa(A,fun(A,A),Times,P2),aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),P2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))) ).

% power.power_eq_if
tff(fact_7228_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A] : aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),A3),zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_7229_power_Opower_Opower__Suc,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),A3),aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),Times,A3),aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),A3),Nb)) ).

% power.power.power_Suc
tff(fact_7230_power_Opower_Ocong,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A))] : power2(A,One,Times) = power2(A,One,Times) ).

% power.power.cong
tff(fact_7231_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat,S: A] :
          ( finite_finite(A,S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),set_ord_lessThan(A,S))))
           => ( infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),set_ord_lessThan(A,S)),Nb) = infini527867602293511546merate(A,S3,Nb) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_7232_butlast__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),butlast(A),take(A,Nb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_7233_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Mb: nat,Nb: nat] :
          ( ~ finite_finite(A,S3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Mb)),infini527867602293511546merate(A,S3,Nb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% enumerate_mono_iff
tff(fact_7234_length__butlast,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(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_butlast
tff(fact_7235_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Mb: nat,Nb: nat] :
          ( finite_finite(A,S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(set(A),nat,finite_card(A),S3))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),S3))
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Mb)),infini527867602293511546merate(A,S3,Nb))
              <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_7236_le__enumerate,axiom,
    ! [S3: set(nat),Nb: nat] :
      ( ~ finite_finite(nat,S3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),infini527867602293511546merate(nat,S3,Nb)) ) ).

% le_enumerate
tff(fact_7237_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( ~ finite_finite(A,S3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Nb)),infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb))) ) ) ).

% enumerate_step
tff(fact_7238_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Mb: nat,Nb: nat,S3: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( ~ finite_finite(A,S3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Mb)),infini527867602293511546merate(A,S3,Nb)) ) ) ) ).

% enumerate_mono
tff(fact_7239_nth__butlast,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),butlast(A),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),butlast(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),Nb) ) ) ).

% nth_butlast
tff(fact_7240_sorted__butlast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),butlast(A),Xs)) ) ) ) ).

% sorted_butlast
tff(fact_7241_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( finite_finite(A,S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),S3))
           => member(A,infini527867602293511546merate(A,S3,Nb),S3) ) ) ) ).

% finite_enumerate_in_set
tff(fact_7242_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),S: A] :
          ( finite_finite(A,S3)
         => ( member(A,S,S3)
           => ? [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(set(A),nat,finite_card(A),S3))
                & ( infini527867602293511546merate(A,S3,N) = S ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_7243_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y6: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X6))
             => ( infini527867602293511546merate(A,X6,I2) = infini527867602293511546merate(A,Y6,I2) ) )
         => ( finite_finite(A,X6)
           => ( finite_finite(A,Y6)
             => ( ( aa(set(A),nat,finite_card(A),X6) = aa(set(A),nat,finite_card(A),Y6) )
               => ( X6 = Y6 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_7244_take__butlast,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))
     => ( take(A,Nb,aa(list(A),list(A),butlast(A),Xs)) = take(A,Nb,Xs) ) ) ).

% take_butlast
tff(fact_7245_butlast__power,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),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),butlast(A)),Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs) ).

% butlast_power
tff(fact_7246_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Mb: nat,Nb: nat,S3: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( finite_finite(A,S3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),S3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Mb)),infini527867602293511546merate(A,S3,Nb)) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_7247_butlast__conv__take,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xs) ).

% butlast_conv_take
tff(fact_7248_butlast__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xb: A] :
      aa(list(A),list(A),butlast(A),list_update(A,Xs,K,Xb)) = $ite(K = 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),list(A),butlast(A),Xs),list_update(A,aa(list(A),list(A),butlast(A),Xs),K,Xb)) ).

% butlast_list_update
tff(fact_7249_finite__le__enumerate,axiom,
    ! [S3: set(nat),Nb: nat] :
      ( finite_finite(nat,S3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(nat),nat,finite_card(nat),S3))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),infini527867602293511546merate(nat,S3,Nb)) ) ) ).

% finite_le_enumerate
tff(fact_7250_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( finite_finite(A,S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),S3))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Nb)),infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb))) ) ) ) ).

% finite_enumerate_step
tff(fact_7251_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y6: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X6))
             => ( infini527867602293511546merate(A,X6,I2) = infini527867602293511546merate(A,Y6,I2) ) )
         => ( finite_finite(A,X6)
           => ( finite_finite(A,Y6)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X6)),aa(set(A),nat,finite_card(A),Y6))
               => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),Y6) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_7252_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( finite_finite(A,S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),S3))
           => ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_acu(set(A),fun(nat,fun(A,$o)),S3),Nb)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_7253_DeMoivre2,axiom,
    ! [R: real,A3: real,Nb: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),rcis(R,A3)),Nb) = rcis(aa(nat,real,aa(real,fun(nat,real),power_power(real),R),Nb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A3)) ).

% DeMoivre2
tff(fact_7254_Least__eq__0,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ord_Least(nat,P) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_7255_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),K) ) ) ).

% Least_le
tff(fact_7256_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] :
          ( ? [X5: A] :
              ( aa(A,$o,P,X5)
              & ! [Y4: A] :
                  ( aa(A,$o,P,Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Y4) )
              & ! [Y4: A] :
                  ( ( aa(A,$o,P,Y4)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Ya2) ) )
                 => ( Y4 = X5 ) ) )
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% Least1I
tff(fact_7257_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Z: A] :
          ( ? [X5: A] :
              ( aa(A,$o,P,X5)
              & ! [Y4: A] :
                  ( aa(A,$o,P,Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Y4) )
              & ! [Y4: A] :
                  ( ( aa(A,$o,P,Y4)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Ya2) ) )
                 => ( Y4 = X5 ) ) )
         => ( aa(A,$o,P,Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),Z) ) ) ) ).

% Least1_le
tff(fact_7258_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y4) )
           => ( ! [X4: A] :
                  ( aa(A,$o,P,X4)
                 => ( ! [Y5: A] :
                        ( aa(A,$o,P,Y5)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y5) )
                   => aa(A,$o,Q,X4) ) )
             => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ) ).

% LeastI2_order
tff(fact_7259_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y4) )
           => ( ord_Least(A,P) = Xb ) ) ) ) ).

% Least_equality
tff(fact_7260_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A3)
         => ( ! [A5: A] :
                ( aa(A,$o,P,A5)
               => ( ! [B10: A] :
                      ( aa(A,$o,P,B10)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B10) )
                 => aa(A,$o,Q,A5) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder
tff(fact_7261_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P,X_13)
         => ( ! [A5: A] :
                ( aa(A,$o,P,A5)
               => ( ! [B10: A] :
                      ( aa(A,$o,P,B10)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B10) )
                 => aa(A,$o,Q,A5) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_7262_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),ord_Least(A,P))
         => ~ aa(A,$o,P,K) ) ) ).

% not_less_Least
tff(fact_7263_Least__Suc2,axiom,
    ! [P: fun(nat,$o),Nb: nat,Q: fun(nat,$o),Mb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,Q,Mb)
       => ( ~ aa(nat,$o,P,zero_zero(nat))
         => ( ! [K3: nat] :
                ( aa(nat,$o,P,aa(nat,nat,suc,K3))
              <=> aa(nat,$o,Q,K3) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_7264_Least__Suc,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_acv(fun(nat,$o),fun(nat,$o),P))) ) ) ) ).

% Least_Suc
tff(fact_7265_Sup__real__def,axiom,
    ! [X6: set(real)] : aa(set(real),real,complete_Sup_Sup(real),X6) = ord_Least(real,aTP_Lamp_acw(set(real),fun(real,$o),X6)) ).

% Sup_real_def
tff(fact_7266_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),S3: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( ? [X5: A] :
                ( member(A,X5,S3)
                & ! [Xa3: A] :
                    ( member(A,Xa3,S3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Xa3) ) )
           => ( ord_Least(B,aa(set(A),fun(B,$o),aTP_Lamp_acx(fun(A,B),fun(set(A),fun(B,$o)),F2),S3)) = aa(A,B,F2,ord_Least(A,aTP_Lamp_acy(set(A),fun(A,$o),S3))) ) ) ) ) ).

% Least_mono
tff(fact_7267_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,$o)] : ord_Least(A,P) = the(A,aTP_Lamp_acz(fun(A,$o),fun(A,$o),P)) ) ).

% Least_def
tff(fact_7268_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( ~ finite_finite(A,S3)
         => ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_acu(set(A),fun(nat,fun(A,$o)),S3),Nb)) ) ) ) ).

% enumerate_Suc''
tff(fact_7269_lexn_Osimps_I2_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Nb: nat] : lexn(A,R,aa(nat,nat,suc,Nb)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),inf_inf(set(product_prod(list(A),list(A)))),image(product_prod(product_prod(A,list(A)),product_prod(A,list(A))),product_prod(list(A),list(A)),product_map_prod(product_prod(A,list(A)),list(A),product_prod(A,list(A)),list(A),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A))),lex_prod(A,list(A),R,lexn(A,R,Nb)))),aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_ada(nat,fun(list(A),fun(list(A),$o)),Nb)))) ).

% lexn.simps(2)
tff(fact_7270_tendsto__iff__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E6: fun(product_prod(B,B),$o)] :
              ( eventually(product_prod(B,B),E6,topolo7806501430040627800ormity(B))
             => eventually(A,aa(fun(product_prod(B,B),$o),fun(A,$o),aa(B,fun(fun(product_prod(B,B),$o),fun(A,$o)),aTP_Lamp_adb(fun(A,B),fun(B,fun(fun(product_prod(B,B),$o),fun(A,$o))),F2),L),E6),F4) ) ) ) ).

% tendsto_iff_uniformity
tff(fact_7271_map__prod__simp,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(C,A),G: fun(D,B),A3: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F2,G),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A3),B2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,A3)),aa(D,B,G,B2)) ).

% map_prod_simp
tff(fact_7272_map__prod__imageI,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B)),F2: fun(A,C),G: fun(B,D)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R2)
     => member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,F2,A3)),aa(B,D,G,B2)),image(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F2,G),R2)) ) ).

% map_prod_imageI
tff(fact_7273_prod__fun__imageE,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod(A,B),F2: fun(C,A),G: fun(D,B),R2: set(product_prod(C,D))] :
      ( member(product_prod(A,B),C2,image(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F2,G),R2))
     => ~ ! [X4: C,Y4: D] :
            ( ( C2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,X4)),aa(D,B,G,Y4)) )
           => ~ member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X4),Y4),R2) ) ) ).

% prod_fun_imageE
tff(fact_7274_map__prod__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F2: fun(A,C),G: fun(B,D)] : product_map_prod(A,C,B,D,F2,G) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_adc(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F2),G)) ).

% map_prod_def
tff(fact_7275_uniformity__transE,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ~ ! [D8: fun(product_prod(A,A),$o)] :
                ( eventually(product_prod(A,A),D8,topolo7806501430040627800ormity(A))
               => ~ ! [X5: A,Y5: A] :
                      ( aa(product_prod(A,A),$o,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y5))
                     => ! [Z4: A] :
                          ( aa(product_prod(A,A),$o,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z4))
                         => aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Z4)) ) ) ) ) ) ).

% uniformity_transE
tff(fact_7276_uniformity__trans,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ? [D8: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),D8,topolo7806501430040627800ormity(A))
              & ! [X5: A,Y5: A,Z4: A] :
                  ( aa(product_prod(A,A),$o,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y5))
                 => ( aa(product_prod(A,A),$o,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z4))
                   => aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Z4)) ) ) ) ) ) ).

% uniformity_trans
tff(fact_7277_uniformity__refl,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o),Xb: A] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Xb)) ) ) ).

% uniformity_refl
tff(fact_7278_uniformity__sym,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(A,A),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_add(fun(product_prod(A,A),$o),fun(A,fun(A,$o)),E5)),topolo7806501430040627800ormity(A)) ) ) ).

% uniformity_sym
tff(fact_7279_Cauchy__uniform__iff,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [P5: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),P5,topolo7806501430040627800ormity(A))
             => ? [N6: nat] :
                ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N4)
                 => ! [M5: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M5)
                     => aa(product_prod(A,A),$o,P5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,X6,N4)),aa(nat,A,X6,M5))) ) ) ) ) ) ).

% Cauchy_uniform_iff
tff(fact_7280_totally__bounded__def,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S3: set(A)] :
          ( topolo6688025880775521714ounded(A,S3)
        <=> ! [E6: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),E6,topolo7806501430040627800ormity(A))
             => ? [X7: set(A)] :
                  ( finite_finite(A,X7)
                  & ! [X: A] :
                      ( member(A,X,S3)
                     => ? [Xa2: A] :
                          ( member(A,Xa2,X7)
                          & aa(product_prod(A,A),$o,E6,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X)) ) ) ) ) ) ) ).

% totally_bounded_def
tff(fact_7281_eventually__uniformity__metric,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [P: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),P,topolo7806501430040627800ormity(A))
        <=> ? [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
              & ! [X: A,Y3: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y3)),E4)
                 => aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)) ) ) ) ) ).

% eventually_uniformity_metric
tff(fact_7282_uniformity__trans_H,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(product_prod(A,A),product_prod(A,A)),aa(fun(product_prod(A,A),fun(product_prod(A,A),$o)),fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),product_case_prod(product_prod(A,A),product_prod(A,A),$o),aa(fun(A,fun(A,fun(product_prod(A,A),$o))),fun(product_prod(A,A),fun(product_prod(A,A),$o)),product_case_prod(A,A,fun(product_prod(A,A),$o)),aTP_Lamp_adf(fun(product_prod(A,A),$o),fun(A,fun(A,fun(product_prod(A,A),$o))),E5))),prod_filter(product_prod(A,A),product_prod(A,A),topolo7806501430040627800ormity(A),topolo7806501430040627800ormity(A))) ) ) ).

% uniformity_trans'
tff(fact_7283_uniformly__continuous__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S: set(A),F2: fun(A,B),E5: fun(product_prod(B,B),$o)] :
          ( topolo6026614971017936543ous_on(A,B,S,F2)
         => ( eventually(product_prod(B,B),E5,topolo7806501430040627800ormity(B))
           => eventually(product_prod(A,A),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(product_prod(B,B),$o),fun(A,fun(A,$o)),aa(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o))),aTP_Lamp_adg(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o)))),S),F2),E5)),topolo7806501430040627800ormity(A)) ) ) ) ).

% uniformly_continuous_onD
tff(fact_7284_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,P: fun(product_prod(A,B),$o),F4: filter(A),G6: filter(B)] :
      ( eventually(product_prod(A,B),P,prod_filter(A,B,F4,G6))
    <=> ? [Pf: fun(A,$o),Pg: fun(B,$o)] :
          ( eventually(A,Pf,F4)
          & eventually(B,Pg,G6)
          & ! [X: A,Y3: B] :
              ( aa(A,$o,Pf,X)
             => ( aa(B,$o,Pg,Y3)
               => aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)) ) ) ) ) ).

% eventually_prod_filter
tff(fact_7285_eventually__prod__same,axiom,
    ! [A: $tType,P: fun(product_prod(A,A),$o),F4: filter(A)] :
      ( eventually(product_prod(A,A),P,prod_filter(A,A,F4,F4))
    <=> ? [Q7: fun(A,$o)] :
          ( eventually(A,Q7,F4)
          & ! [X: A,Y3: A] :
              ( aa(A,$o,Q7,X)
             => ( aa(A,$o,Q7,Y3)
               => aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)) ) ) ) ) ).

% eventually_prod_same
tff(fact_7286_nhds__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [A3: A,B2: B] : topolo7230453075368039082e_nhds(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) = prod_filter(A,B,topolo7230453075368039082e_nhds(A,A3),topolo7230453075368039082e_nhds(B,B2)) ) ).

% nhds_prod
tff(fact_7287_filterlim__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G6: filter(B),F4: filter(A),G: fun(A,C),H6: filter(C)] :
      ( filterlim(A,B,F2,G6,F4)
     => ( filterlim(A,C,G,H6,F4)
       => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_adh(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),prod_filter(B,C,G6,H6),F4) ) ) ).

% filterlim_Pair
tff(fact_7288_uniformly__continuous__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S,F2)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X: A] :
                      ( member(A,X,S)
                     => ! [Xa2: A] :
                          ( member(A,Xa2,S)
                         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa2,X)),D4)
                           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,Xa2),aa(A,B,F2,X))),E4) ) ) ) ) ) ) ) ).

% uniformly_continuous_on_def
tff(fact_7289_tendsto__mult__Pair,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [A3: A,B2: A] : filterlim(product_prod(A,A),A,aTP_Lamp_adi(product_prod(A,A),A),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),prod_filter(A,A,topolo7230453075368039082e_nhds(A,A3),topolo7230453075368039082e_nhds(A,B2))) ) ).

% tendsto_mult_Pair
tff(fact_7290_tendsto__add__Pair,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [A3: A,B2: A] : filterlim(product_prod(A,A),A,aTP_Lamp_adj(product_prod(A,A),A),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),prod_filter(A,A,topolo7230453075368039082e_nhds(A,A3),topolo7230453075368039082e_nhds(A,B2))) ) ).

% tendsto_add_Pair
tff(fact_7291_isUCont__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,top_top(set(A)),F2)
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X: A,Y3: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y3)),S6)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X),aa(A,B,F2,Y3))),R5) ) ) ) ) ) ).

% isUCont_def
tff(fact_7292_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: filter(A),G6: filter(B),H6: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F4,G6),H6) = filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),product_case_prod(A,product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_adl(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)))),prod_filter(A,product_prod(B,C),F4,prod_filter(B,C,G6,H6))) ).

% prod_filter_assoc
tff(fact_7293_uniformity__dist,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ( topolo7806501430040627800ormity(A) = aa(set(filter(product_prod(A,A))),filter(product_prod(A,A)),complete_Inf_Inf(filter(product_prod(A,A))),image(real,filter(product_prod(A,A)),aTP_Lamp_adn(real,filter(product_prod(A,A))),set_ord_greaterThan(real,zero_zero(real)))) ) ) ).

% uniformity_dist
tff(fact_7294_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),Xb: B] : prod_filter(A,B,F4,principal(B,aa(set(B),set(B),insert(B,Xb),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_nl(B,fun(A,product_prod(A,B))),Xb),F4) ).

% prod_filter_principal_singleton2
tff(fact_7295_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,Xb: A,F4: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))),F4) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),F4) ).

% prod_filter_principal_singleton
tff(fact_7296_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),F4: filter(C)] : aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),filtermap(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_ado(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G),F4)),prod_filter(A,B,filtermap(C,A,F2,F4),filtermap(C,B,G,F4))) ).

% filtermap_Pair
tff(fact_7297_filtermap__nhds__times,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C2),topolo7230453075368039082e_nhds(A,A3)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) ) ) ) ).

% filtermap_nhds_times
tff(fact_7298_uniformity__complex__def,axiom,
    topolo7806501430040627800ormity(complex) = aa(set(filter(product_prod(complex,complex))),filter(product_prod(complex,complex)),complete_Inf_Inf(filter(product_prod(complex,complex))),image(real,filter(product_prod(complex,complex)),aTP_Lamp_adq(real,filter(product_prod(complex,complex))),set_ord_greaterThan(real,zero_zero(real)))) ).

% uniformity_complex_def
tff(fact_7299_uniformity__real__def,axiom,
    topolo7806501430040627800ormity(real) = aa(set(filter(product_prod(real,real))),filter(product_prod(real,real)),complete_Inf_Inf(filter(product_prod(real,real))),image(real,filter(product_prod(real,real)),aTP_Lamp_ads(real,filter(product_prod(real,real))),set_ord_greaterThan(real,zero_zero(real)))) ).

% uniformity_real_def
tff(fact_7300_eventually__prod__sequentially,axiom,
    ! [P: fun(product_prod(nat,nat),$o)] :
      ( eventually(product_prod(nat,nat),P,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N6: nat] :
        ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M5)
         => ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N4)
             => aa(product_prod(nat,nat),$o,P,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N4),M5)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_7301_at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A] : topolo174197925503356063within(A,A3,top_top(set(A))) = filtermap(A,A,aTP_Lamp_adt(A,fun(A,A),A3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% at_to_0
tff(fact_7302_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_adu(A,fun(A,filter(A)),Xb),set_ord_lessThan(A,Xb))) ) ) ) ).

% at_left_eq
tff(fact_7303_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_adv(A,fun(A,filter(A)),Xb),set_ord_greaterThan(A,Xb))) ) ) ) ).

% at_right_eq
tff(fact_7304_filtermap__times__pos__at__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [C2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C2),topolo174197925503356063within(A,P2,set_ord_greaterThan(A,P2))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2),set_ord_greaterThan(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2))) ) ) ) ).

% filtermap_times_pos_at_right
tff(fact_7305_cauchy__filter__metric__filtermap,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F2: fun(B,A),F4: filter(B)] :
          ( topolo6773858410816713723filter(A,filtermap(B,A,F2,F4))
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [P5: fun(B,$o)] :
                  ( eventually(B,P5,F4)
                  & ! [X: B,Y3: B] :
                      ( ( aa(B,$o,P5,X)
                        & aa(B,$o,P5,Y3) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(B,A,F2,X),aa(B,A,F2,Y3))),E4) ) ) ) ) ) ).

% cauchy_filter_metric_filtermap
tff(fact_7306_nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A] : topolo7230453075368039082e_nhds(A,Xb) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(real,filter(A),aTP_Lamp_adx(A,fun(real,filter(A)),Xb),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% nhds_metric
tff(fact_7307_fold__atLeastAtMost__nat_Opelims,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 )
     => ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),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))),Xb),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)),Xaa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xba),Xc))))
       => ~ ( ( 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))) )
           => ~ accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),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))),Xb),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)),Xaa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xba),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_7308_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),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)),A3),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2))))
     => ( set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A3),Acc2,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A3),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_7309_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),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))),A0),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)),A1),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A22),A32))))
     => ( ! [F3: fun(nat,fun(A,A)),A5: nat,B3: nat,Acc: A] :
            ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A5),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),A5)
               => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),one_one(nat))),B3),aa(A,A,aa(nat,fun(A,A),F3,A5),Acc)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F3),A5),B3),Acc) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_7310_Gcd__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = $ite(finite_finite(A,A4),finite_fold(A,A,gcd_gcd(A),zero_zero(A),A4),one_one(A)) ) ).

% Gcd_fin.eq_fold
tff(fact_7311_uniformly__continuous__on__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S,F2)
        <=> filterlim(product_prod(A,A),product_prod(B,B),aa(fun(A,fun(A,product_prod(B,B))),fun(product_prod(A,A),product_prod(B,B)),product_case_prod(A,A,product_prod(B,B)),aTP_Lamp_ady(fun(A,B),fun(A,fun(A,product_prod(B,B))),F2)),topolo7806501430040627800ormity(B),aa(filter(product_prod(A,A)),filter(product_prod(A,A)),aa(filter(product_prod(A,A)),fun(filter(product_prod(A,A)),filter(product_prod(A,A))),inf_inf(filter(product_prod(A,A))),topolo7806501430040627800ormity(A)),principal(product_prod(A,A),product_Sigma(A,A,S,aTP_Lamp_adz(set(A),fun(A,set(A)),S))))) ) ) ).

% uniformly_continuous_on_uniformity
tff(fact_7312_SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: A,A4: set(A),B2: B,B4: fun(A,set(B))] :
      ( member(A,A3,A4)
     => ( member(B,B2,aa(A,set(B),B4,A3))
       => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,A4,B4)) ) ) ).

% SigmaI
tff(fact_7313_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B4: 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),A3),B2),product_Sigma(A,B,A4,B4))
    <=> ( member(A,A3,A4)
        & member(B,B2,aa(A,set(B),B4,A3)) ) ) ).

% mem_Sigma_iff
tff(fact_7314_Gcd__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ~ finite_finite(A,A4)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).

% Gcd_fin.infinite
tff(fact_7315_is__unit__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( dvd_dvd(A,aa(set(A),A,semiring_gcd_Gcd_fin(A),A4),one_one(A))
        <=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).

% is_unit_Gcd_fin_iff
tff(fact_7316_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A3: A,A4: set(A),B2: B,B4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),insert(A,A3),A4),aa(set(B),fun(A,set(B)),aTP_Lamp_aea(B,fun(set(B),fun(A,set(B))),B2),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),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,A4,aa(set(B),fun(A,set(B)),aTP_Lamp_aea(B,fun(set(B),fun(A,set(B))),B2),B4))),product_Sigma(A,B,aa(set(A),set(A),insert(A,A3),A4),aTP_Lamp_aeb(set(B),fun(A,set(B)),B4)))) ).

% insert_Times_insert
tff(fact_7317_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B4: 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),A3),B2),product_Sigma(A,B,A4,B4))
     => ~ ( member(A,A3,A4)
         => ~ member(B,B2,aa(A,set(B),B4,A3)) ) ) ).

% SigmaE2
tff(fact_7318_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B4: 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),A3),B2),product_Sigma(A,B,A4,B4))
     => member(B,B2,aa(A,set(B),B4,A3)) ) ).

% SigmaD2
tff(fact_7319_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B4: 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),A3),B2),product_Sigma(A,B,A4,B4))
     => member(A,A3,A4) ) ).

% SigmaD1
tff(fact_7320_SigmaE,axiom,
    ! [A: $tType,B: $tType,C2: product_prod(A,B),A4: set(A),B4: fun(A,set(B))] :
      ( member(product_prod(A,B),C2,product_Sigma(A,B,A4,B4))
     => ~ ! [X4: A] :
            ( member(A,X4,A4)
           => ! [Y4: B] :
                ( member(B,Y4,aa(A,set(B),B4,X4))
               => ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) ) ) ) ) ).

% SigmaE
tff(fact_7321_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),A4: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_aec(set(A),fun(A,set(A)),A4)))
       => ( ( A3 = B2 )
          | member(A,A3,A4) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_7322_wfI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_aec(set(A),fun(A,set(A)),B4)))
     => ( ! [X4: A,P6: fun(A,$o)] :
            ( ! [Xa: 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),Xa),R)
                   => aa(A,$o,P6,Y4) )
               => aa(A,$o,P6,Xa) )
           => ( member(A,X4,A4)
             => ( member(A,X4,B4)
               => aa(A,$o,P6,X4) ) ) )
       => wf(A,R) ) ) ).

% wfI
tff(fact_7323_swap__product,axiom,
    ! [A: $tType,B: $tType,A4: set(B),B4: 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_nl(B,fun(A,product_prod(A,B)))),product_Sigma(B,A,A4,aTP_Lamp_aed(set(A),fun(B,set(A)),B4))) = product_Sigma(A,B,B4,aTP_Lamp_aeb(set(B),fun(A,set(B)),A4)) ).

% swap_product
tff(fact_7324_card__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B4: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_aeb(set(B),fun(A,set(B)),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B4)) ).

% card_cartesian_product
tff(fact_7325_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F2: fun(C,A),G: fun(D,B),A4: set(C),B4: 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_mx(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F2),G)),product_Sigma(C,D,A4,aTP_Lamp_aee(set(D),fun(C,set(D)),B4))) = product_Sigma(A,B,image(C,A,F2,A4),aa(set(D),fun(A,set(B)),aTP_Lamp_aef(fun(D,B),fun(set(D),fun(A,set(B))),G),B4)) ).

% image_paired_Times
tff(fact_7326_lists__length__Suc__eq,axiom,
    ! [A: $tType,A4: set(A),Nb: nat] : aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_aeg(set(A),fun(nat,fun(list(A),$o)),A4),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_aeh(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_ah(set(A),fun(nat,fun(list(A),$o)),A4),Nb)),aTP_Lamp_aei(set(A),fun(list(A),set(A)),A4))) ).

% lists_length_Suc_eq
tff(fact_7327_mult__inj__if__coprime__nat,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),A4: set(A),G: fun(B,nat),B4: set(B)] :
      ( inj_on(A,nat,F2,A4)
     => ( inj_on(B,nat,G,B4)
       => ( ! [A5: A,B3: B] :
              ( member(A,A5,A4)
             => ( member(B,B3,B4)
               => algebr8660921524188924756oprime(nat,aa(A,nat,F2,A5),aa(B,nat,G,B3)) ) )
         => 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_aej(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F2),G)),product_Sigma(A,B,A4,aTP_Lamp_aeb(set(B),fun(A,set(B)),B4))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_7328_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_gj(nat,fun(nat,fun(nat,$o)),Mb))) = product_Sigma(nat,nat,set_ord_atMost(nat,Mb),aTP_Lamp_aek(nat,fun(nat,set(nat)),Mb)) ).

% pairs_le_eq_Sigma
tff(fact_7329_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B4: fun(A,set(B))] : product_Sigma(A,B,A4,B4) = 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_aem(fun(A,set(B)),fun(A,set(product_prod(A,B))),B4),A4)) ).

% Sigma_def
tff(fact_7330_product__fold,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B4: set(B)] :
      ( finite_finite(A,A4)
     => ( finite_finite(B,B4)
       => ( product_Sigma(A,B,A4,aTP_Lamp_aeb(set(B),fun(A,set(B)),B4)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_aeo(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B4),bot_bot(set(product_prod(A,B))),A4) ) ) ) ).

% product_fold
tff(fact_7331_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),A4: set(C)] : image(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_ado(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G),A4) = product_Sigma(A,B,image(C,A,F2,A4),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_aep(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F2),G),A4)) ).

% image_split_eq_Sigma
tff(fact_7332_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_0(C)
     => ! [F2: fun(A,C),G: fun(B,C),A4: set(A),B4: set(B)] :
          ( inj_on(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_aeq(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),F2),G)),product_Sigma(A,B,A4,aTP_Lamp_aeb(set(B),fun(A,set(B)),B4)))
         => ( aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(A),C,groups7311177749621191930dd_sum(A,C,F2),A4)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),B4)) = aa(set(C),C,groups7311177749621191930dd_sum(C,C,id(C)),aa(fun(C,$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,$o))),aTP_Lamp_aer(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),F2),G),A4),B4))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_7333_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_7334_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_7335_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_7336_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_7337_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_7338_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_7339_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,Xb: B,A4: set(B),F2: 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,A4,F2)) = $ite(member(B,Xb,A4),aa(B,set(A),F2,Xb),bot_bot(set(A))) ).

% Pair_vimage_Sigma
tff(fact_7340_map__option_Oidentity,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_mf(A,A)) = id(option(A)) ).

% map_option.identity
tff(fact_7341_option_Omap__id0,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),id(A)) = id(option(A)) ).

% option.map_id0
tff(fact_7342_option_Omap__id,axiom,
    ! [A: $tType,Ta: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),id(A)),Ta) = Ta ).

% option.map_id
tff(fact_7343_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F2: 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)),F2) = id(A) ).

% funpow_simps_right(1)
tff(fact_7344_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_nh(A,product_prod(A,A))),Z) = aa(A,A,id(A),Z) ).

% fst_diag_id
tff(fact_7345_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_nh(A,product_prod(A,A))),Z) = aa(A,A,id(A),Z) ).

% snd_diag_id
tff(fact_7346_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),D5: set(A),A4: set(B)] :
      ( inj_on(A,B,F2,D5)
     => ( finite_finite(B,A4)
       => aa(nat,$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,F2,A4)),D5))),aa(set(B),nat,finite_card(B),A4)) ) ) ).

% card_vimage_inj_on_le
tff(fact_7347_set__decode__div__2,axiom,
    ! [Xb: nat] : nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = vimage(nat,nat,suc,nat_set_decode(Xb)) ).

% set_decode_div_2
tff(fact_7348_set__encode__vimage__Suc,axiom,
    ! [A4: set(nat)] : aa(set(nat),nat,nat_set_encode,vimage(nat,nat,suc,A4)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(set(nat),nat,nat_set_encode,A4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% set_encode_vimage_Suc
tff(fact_7349_Real_Opositive__def,axiom,
    positive2 = aa(fun(fun(nat,rat),$o),fun(real,$o),map_fun(real,fun(nat,rat),$o,$o,rep_real2,id($o)),aTP_Lamp_ace(fun(nat,rat),$o)) ).

% Real.positive_def
tff(fact_7350_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_aci(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_7351_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($o)),aTP_Lamp_acg(product_prod(int,int),$o)) ).

% Rat.positive_def
tff(fact_7352_Restr__natLeq,axiom,
    ! [Nb: nat] : 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))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),Nb)),aTP_Lamp_aes(nat,fun(nat,set(nat)),Nb))) = aa(fun(product_prod(nat,nat),$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,$o),aTP_Lamp_abz(nat,fun(nat,fun(nat,$o)),Nb))) ).

% Restr_natLeq
tff(fact_7353_relation__of__def,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),A4: set(A)] : order_relation_of(A,P,A4) = 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(A),fun(A,fun(A,$o)),aTP_Lamp_aet(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),P),A4))) ).

% relation_of_def
tff(fact_7354_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = 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_eq(nat))) ).

% natLeq_def
tff(fact_7355_Restr__natLeq2,axiom,
    ! [Nb: nat] : 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))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,Nb),aTP_Lamp_aeu(nat,fun(nat,set(nat)),Nb))) = 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_abz(nat,fun(nat,fun(nat,$o)),Nb))) ).

% Restr_natLeq2
tff(fact_7356_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),Nb: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,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) ) ) ) ).

% possible_bit_def
tff(fact_7357_possible__bit__min,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J))
        <=> ( bit_se6407376104438227557le_bit(A,Tyrep,I)
            | bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_min
tff(fact_7358_natLeq__underS__less,axiom,
    ! [Nb: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,Nb) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),Nb)) ).

% natLeq_underS_less
tff(fact_7359_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
           => bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_7360_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat)) ) ).

% possible_bit_0
tff(fact_7361_underS__Field2,axiom,
    ! [A: $tType,A3: A,R: set(product_prod(A,A))] :
      ( member(A,A3,field2(A,R))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R,A3)),field2(A,R)) ) ).

% underS_Field2
tff(fact_7362_underS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_underS(A,R,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aev(set(product_prod(A,A)),fun(A,fun(A,$o)),R),A3)) ).

% underS_def
tff(fact_7363_underS__Field3,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( ( field2(A,R) != bot_bot(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R,A3)),field2(A,R)) ) ).

% underS_Field3
tff(fact_7364_underS__incl__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( member(A,A3,field2(A,R))
       => ( member(A,B2,field2(A,R))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R,A3)),order_underS(A,R,B2))
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R) ) ) ) ) ).

% underS_incl_iff
tff(fact_7365_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] :
          aa(A,A,bit_se4197421643247451524op_bit(A,Mb),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
                & bit_se6407376104438227557le_bit(A,type2(A),Nb) ))),
            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))) ) ).

% drop_bit_exp_eq
tff(fact_7366_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% bit_minus_2_iff
tff(fact_7367_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),one_one(A))),Nb)
        <=> bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ).

% bit_minus_1_iff
tff(fact_7368_bit__eqI,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A] :
          ( ! [N: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) ) )
         => ( A3 = B2 ) ) ) ).

% bit_eqI
tff(fact_7369_bit__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ! [N4: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N4)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N4)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N4) ) ) ) ) ).

% bit_eq_iff
tff(fact_7370_impossible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,A3: A] :
          ( ~ bit_se6407376104438227557le_bit(A,type2(A),Nb)
         => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb) ) ) ).

% impossible_bit
tff(fact_7371_bit__imp__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
         => bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ).

% bit_imp_possible_bit
tff(fact_7372_bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb) ) ) ) ).

% bit_not_iff
tff(fact_7373_bit__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Mb),A3)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb)
            | ( ( Mb = Nb )
              & bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ) ) ).

% bit_set_bit_iff
tff(fact_7374_bit__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,bit_se8732182000553998342ip_bit(A,Mb,A3)),Nb)
        <=> ( ( ( Mb = Nb )
            <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Nb) )
            & bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ) ).

% bit_flip_bit_iff
tff(fact_7375_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,Mb)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ).

% bit_mask_iff
tff(fact_7376_bit__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(int,A,ring_1_of_int(A),K)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ) ) ).

% bit_of_int_iff
tff(fact_7377_bit__of__nat__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),Mb)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Mb),Nb) ) ) ) ).

% bit_of_nat_iff
tff(fact_7378_bit__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4674362597316999326ke_bit(A,Mb),A3)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)) ) ) ) ).

% bit_signed_take_bit_iff
tff(fact_7379_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A3)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),Nb) ) ) ) ).

% bit_minus_iff
tff(fact_7380_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A3: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),A3)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
            & bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ) ).

% bit_push_bit_iff
tff(fact_7381_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: 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) )
        <=> ~ bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ).

% fold_possible_bit
tff(fact_7382_bit__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ( Mb = Nb ) ) ) ) ).

% bit_exp_iff
tff(fact_7383_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),one_one(nat))
            & ( Nb = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_7384_bit__not__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ( Nb != Mb ) ) ) ) ).

% bit_not_exp_iff
tff(fact_7385_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7386_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(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))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7387_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Nb: nat] :
          ( aa(nat,$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))),A3)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))
            & ( Nb != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ) ).

% bit_double_iff
tff(fact_7388_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
            & ( aa(nat,A,semiring_1_of_nat(A),N4) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7389_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X4)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),C2)
                   => ( aa(nat,A,semiring_1_of_nat(A),X4) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7390_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
           => ( aa(nat,A,semiring_1_of_nat(A),N4) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7391_underS__E,axiom,
    ! [A: $tType,I: A,R2: set(product_prod(A,A)),J: A] :
      ( member(A,I,order_underS(A,R2,J))
     => ( ( I != J )
        & 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),R2) ) ) ).

% underS_E
tff(fact_7392_underS__I,axiom,
    ! [A: $tType,I: A,J: A,R2: set(product_prod(A,A))] :
      ( ( I != J )
     => ( 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),R2)
       => member(A,I,order_underS(A,R2,J)) ) ) ).

% underS_I
tff(fact_7393_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R: A,S: B,R2: set(product_prod(A,B)),S9: B] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S),R2)
     => ( ( S9 = S )
       => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S9),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_7394_acyclic__insert,axiom,
    ! [A: $tType,Y: A,Xb: A,R: set(product_prod(A,A))] :
      ( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xb)),R))
    <=> ( transitive_acyclic(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),Xb),Y),transitive_rtrancl(A,R)) ) ) ).

% acyclic_insert
tff(fact_7395_Rat_Opositive_Orsp,axiom,
    aa(fun(product_prod(int,int),$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),$o,$o,ratrel,fequal($o)),aTP_Lamp_acg(product_prod(int,int),$o)),aTP_Lamp_acg(product_prod(int,int),$o)) ).

% Rat.positive.rsp
tff(fact_7396_acyclicI__order,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [R: set(product_prod(A,A)),F2: fun(A,B)] :
          ( ! [A5: A,B3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B3),R)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B3)),aa(A,B,F2,A5)) )
         => transitive_acyclic(A,R) ) ) ).

% acyclicI_order
tff(fact_7397_acyclic__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( transitive_acyclic(A,R)
    <=> ! [X: 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),X),transitive_trancl(A,R)) ) ).

% acyclic_def
tff(fact_7398_acyclicI,axiom,
    ! [A: $tType,R: set(product_prod(A,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,R))
     => transitive_acyclic(A,R) ) ).

% acyclicI
tff(fact_7399_of__rat_Orsp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(fun(product_prod(int,int),A),$o,aa(fun(product_prod(int,int),A),fun(fun(product_prod(int,int),A),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),A,A,ratrel,fequal(A)),aTP_Lamp_aci(product_prod(int,int),A)),aTP_Lamp_aci(product_prod(int,int),A)) ) ).

% of_rat.rsp
tff(fact_7400_Rat_Opositive_Oabs__eq,axiom,
    ! [Xb: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xb),Xb)
     => ( aa(rat,$o,positive,abs_Rat(Xb))
      <=> aa(int,$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_7401_of__rat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xb: product_prod(int,int)] :
          ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xb),Xb)
         => ( aa(rat,A,field_char_0_of_rat(A),abs_Rat(Xb)) = aa(A,A,aa(A,fun(A,A),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_7402_last__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xb: A] :
      ( ( Xs != nil(A) )
     => ( last(A,list_update(A,Xs,K,Xb)) = $ite(K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xb,last(A,Xs)) ) ) ).

% last_list_update
tff(fact_7403_init__seg__of__def,axiom,
    ! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),$o),aTP_Lamp_aew(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)))) ).

% init_seg_of_def
tff(fact_7404_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( last(nat,upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_7405_last__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))
     => ( last(A,drop(A,Nb,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_7406_last__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
         => ( last(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),last(A,Xs)),last(B,Ys)) ) ) ) ) ).

% last_zip
tff(fact_7407_last__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( last(A,Xs) = 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)),one_one(nat))) ) ) ).

% last_conv_nth
tff(fact_7408_arg__min__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A3: A] :
          ( inj_on(A,B,F2,aa(fun(A,$o),set(A),collect(A),P))
         => ( aa(A,$o,P,A3)
           => ( ! [Y4: A] :
                  ( aa(A,$o,P,Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),aa(A,B,F2,Y4)) )
             => ( lattices_ord_arg_min(A,B,F2,P) = A3 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_7409_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_7410_refl__onD,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,A4,R)
     => ( member(A,A3,A4)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),R) ) ) ).

% refl_onD
tff(fact_7411_refl__onD1,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A)),Xb: A,Y: A] :
      ( refl_on(A,A4,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),R)
       => member(A,Xb,A4) ) ) ).

% refl_onD1
tff(fact_7412_refl__onD2,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A)),Xb: A,Y: A] :
      ( refl_on(A,A4,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),R)
       => member(A,Y,A4) ) ) ).

% refl_onD2
tff(fact_7413_refl__on__domain,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( refl_on(A,A4,R)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R)
       => ( member(A,A3,A4)
          & member(A,B2,A4) ) ) ) ).

% refl_on_domain
tff(fact_7414_arg__minI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [P: fun(A,$o),Xb: A,F2: fun(A,B),Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y4)),aa(A,B,F2,Xb)) )
           => ( ! [X4: A] :
                  ( aa(A,$o,P,X4)
                 => ( ! [Y5: A] :
                        ( aa(A,$o,P,Y5)
                       => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y5)),aa(A,B,F2,X4)) )
                   => aa(A,$o,Q,X4) ) )
             => aa(A,$o,Q,lattices_ord_arg_min(A,B,F2,P)) ) ) ) ) ).

% arg_minI
tff(fact_7415_arg__min__nat__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Mb: fun(A,nat)] :
      ( aa(A,$o,P,Xb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Mb,lattices_ord_arg_min(A,nat,Mb,P))),aa(A,nat,Mb,Xb)) ) ).

% arg_min_nat_le
tff(fact_7416_arg__min__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,Mb: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ( aa(A,$o,P,lattices_ord_arg_min(A,nat,Mb,P))
        & ! [Y5: A] :
            ( aa(A,$o,P,Y5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Mb,lattices_ord_arg_min(A,nat,Mb,P))),aa(A,nat,Mb,Y5)) ) ) ) ).

% arg_min_nat_lemma
tff(fact_7417_arg__min__equality,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [P: fun(A,$o),K: A,F2: fun(A,B)] :
          ( aa(A,$o,P,K)
         => ( ! [X4: A] :
                ( aa(A,$o,P,X4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,K)),aa(A,B,F2,X4)) )
           => ( aa(A,B,F2,lattices_ord_arg_min(A,B,F2,P)) = aa(A,B,F2,K) ) ) ) ) ).

% arg_min_equality
tff(fact_7418_refl__on__def,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( refl_on(A,A4,R)
    <=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_aec(set(A),fun(A,set(A)),A4)))
        & ! [X: A] :
            ( member(A,X,A4)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R) ) ) ) ).

% refl_on_def
tff(fact_7419_refl__onI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_aec(set(A),fun(A,set(A)),A4)))
     => ( ! [X4: A] :
            ( member(A,X4,A4)
           => 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),R) )
       => refl_on(A,A4,R) ) ) ).

% refl_onI
tff(fact_7420_refl__on__def_H,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( refl_on(A,A4,R)
    <=> ( ! [X: product_prod(A,A)] :
            ( member(product_prod(A,A),X,R)
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_aex(set(A),fun(A,fun(A,$o)),A4)),X) )
        & ! [X: A] :
            ( member(A,X,A4)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R) ) ) ) ).

% refl_on_def'
tff(fact_7421_in__measures_I2_J,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat),Fs: list(fun(A,nat))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
        | ( ( aa(A,nat,F2,Xb) = aa(A,nat,F2,Y) )
          & 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),measures(A,Fs)) ) ) ) ).

% in_measures(2)
tff(fact_7422_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,$o))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),field2(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),A3),B2),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))),R),id2(A)))
              | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A))) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
         => ( ( ( A3 = B2 )
             => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_7423_in__measures_I1_J,axiom,
    ! [A: $tType,Xb: A,Y: 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),measures(A,nil(fun(A,nat)))) ).

% in_measures(1)
tff(fact_7424_well__order__induct__imp,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( ! [X4: A] :
            ( ! [Y5: A] :
                ( ( ( Y5 != X4 )
                  & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X4),R) )
               => ( member(A,Y5,field2(A,R))
                 => aa(A,$o,P,Y5) ) )
           => ( member(A,X4,field2(A,R))
             => aa(A,$o,P,X4) ) )
       => ( member(A,A3,field2(A,R))
         => aa(A,$o,P,A3) ) ) ) ).

% well_order_induct_imp
tff(fact_7425_wo__rel_Omax2__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = $ite(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R),B2,A3) ) ) ).

% wo_rel.max2_def
tff(fact_7426_wo__rel_Omax2__greater,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( member(A,A3,field2(A,R))
       => ( member(A,B2,field2(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),A3),bNF_We1388413361240627857o_max2(A,R,A3,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),bNF_We1388413361240627857o_max2(A,R,A3,B2)),R) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_7427_wo__rel_Omax2__equals2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( member(A,A3,field2(A,R))
       => ( member(A,B2,field2(A,R))
         => ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = B2 )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_7428_wo__rel_Omax2__equals1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( member(A,A3,field2(A,R))
       => ( member(A,B2,field2(A,R))
         => ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = A3 )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_7429_wo__rel_OTOTALS,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ! [X5: A] :
          ( member(A,X5,field2(A,R))
         => ! [Xa: A] :
              ( member(A,Xa,field2(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),X5),Xa),R)
                | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X5),R) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_7430_wo__rel_Owell__order__induct,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( ! [X4: A] :
            ( ! [Y5: A] :
                ( ( ( Y5 != X4 )
                  & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X4),R) )
               => aa(A,$o,P,Y5) )
           => aa(A,$o,P,X4) )
       => aa(A,$o,P,A3) ) ) ).

% wo_rel.well_order_induct
tff(fact_7431_wo__rel_Ocases__Total,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,$o))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),field2(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),A3),B2),R)
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
         => ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R)
             => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_7432_natLeq__on__wo__rel,axiom,
    ! [Nb: nat] : bNF_Wellorder_wo_rel(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_abz(nat,fun(nat,fun(nat,$o)),Nb)))) ).

% natLeq_on_wo_rel
tff(fact_7433_measures__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,Fs: list(fun(A,nat))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,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),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs))) ) ).

% measures_less
tff(fact_7434_measures__lesseq,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,Fs: list(fun(A,nat))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,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),measures(A,Fs))
       => 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),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs))) ) ) ).

% measures_lesseq
tff(fact_7435_wo__rel_Omax2__greater__among,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( member(A,A3,field2(A,R))
       => ( member(A,B2,field2(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),A3),bNF_We1388413361240627857o_max2(A,R,A3,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),bNF_We1388413361240627857o_max2(A,R,A3,B2)),R)
            & member(A,bNF_We1388413361240627857o_max2(A,R,A3,B2),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_7436_wo__rel_Ominim__least,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),field2(A,R))
       => ( member(A,B2,B4)
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R,B4)),B2),R) ) ) ) ).

% wo_rel.minim_least
tff(fact_7437_wo__rel_Oequals__minim,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B4: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),field2(A,R))
       => ( member(A,A3,B4)
         => ( ! [B3: A] :
                ( member(A,B3,B4)
               => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R) )
           => ( A3 = bNF_We6954850376910717587_minim(A,R,B4) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_7438_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_7439_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_7440_sorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Xb: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F2,linord329482645794927042rt_key(B,A,F2,Xb,Xs))) ) ) ).

% sorted_insort_insert_key
tff(fact_7441_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F2: fun(nat,real),M6: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_aey(fun(nat,real),fun(nat,fun(nat,real)),F2),M6),at_top(nat))
     => ( ! [M2: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),M2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,M2)),aa(nat,real,F2,N)) ) )
       => topolo6863149650580417670ergent(real,F2) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_7442_convergent__mult__const__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ul(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_iff
tff(fact_7443_convergent__mult__const__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_um(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_right_iff
tff(fact_7444_convergent__mult,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [X6: fun(nat,A),Y6: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X6)
         => ( topolo6863149650580417670ergent(A,Y6)
           => topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),X6),Y6)) ) ) ) ).

% convergent_mult
tff(fact_7445_convergent__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),Mb: nat] :
          ( topolo6863149650580417670ergent(A,aa(nat,fun(nat,A),aTP_Lamp_uo(fun(nat,A),fun(nat,fun(nat,A)),F2),Mb))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_ignore_initial_segment
tff(fact_7446_convergent__add__const__right__iff,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( topolo6863149650580417670ergent(A,aa(A,fun(nat,A),aTP_Lamp_afa(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_add_const_right_iff
tff(fact_7447_convergent__add__const__iff,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afb(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_add_const_iff
tff(fact_7448_convergent__add,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [X6: fun(nat,A),Y6: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X6)
         => ( topolo6863149650580417670ergent(A,Y6)
           => topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afc(fun(nat,A),fun(fun(nat,A),fun(nat,A)),X6),Y6)) ) ) ) ).

% convergent_add
tff(fact_7449_lim__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),Xb: A] :
          ( topolo6863149650580417670ergent(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),topolo3827282254853284352ce_Lim(nat,A,at_top(nat),F2)),Xb) ) ) ) ).

% lim_le
tff(fact_7450_Bseq__mono__convergent,axiom,
    ! [X6: fun(nat,real)] :
      ( bfun(nat,real,X6,at_top(nat))
     => ( ! [M2: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,X6,M2)),aa(nat,real,X6,N)) )
       => topolo6863149650580417670ergent(real,X6) ) ) ).

% Bseq_mono_convergent
tff(fact_7451_convergent__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))
       => topolo6863149650580417670ergent(real,aa(real,fun(nat,real),power_power(real),Xb)) ) ) ).

% convergent_realpow
tff(fact_7452_sorted__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Xb: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_ov(A,A),Xb,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_7453_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F2: fun(nat,real),M6: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_aey(fun(nat,real),fun(nat,fun(nat,real)),F2),M6),at_top(nat))
     => ( ! [M2: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),M2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,F2,M2)) ) )
       => topolo6863149650580417670ergent(real,F2) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_7454_horner__sum__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [A4: fun(A,fun(B,$o)),B4: fun(C,fun(D,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A4,zero_zero(A)),zero_zero(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),plus_plus(A)),plus_plus(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),times_times(A)),times_times(B))
             => aa(fun(fun(D,B),fun(B,fun(list(D),B))),$o,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),$o),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B4,A4),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A4,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B4),A4))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B)) ) ) ) ) ).

% horner_sum_transfer
tff(fact_7455_pair__lessI2,axiom,
    ! [A3: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S),Ta)
       => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_7456_pair__less__iff1,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Z)),fun_pair_less)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Z) ) ).

% pair_less_iff1
tff(fact_7457_length__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(fun(list(B),nat),$o,aa(fun(list(A),nat),fun(fun(list(B),nat),$o),bNF_rel_fun(list(A),list(B),nat,nat,list_all2(A,B,A4),fequal(nat)),size_size(list(A))),size_size(list(B))) ).

% length_transfer
tff(fact_7458_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,B,nth(B,Ys),I4)) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_7459_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),P: fun(A,fun(B,$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),A3))
           => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,A3),N)),aa(nat,B,nth(B,B2),N)) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),A3),B2) ) ) ).

% list_all2_all_nthI
tff(fact_7460_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),P2: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P2),aa(list(B),nat,size_size(list(B)),Ys))
       => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,Xs),P2)),aa(nat,B,nth(B,Ys),P2)) ) ) ).

% list_all2_nthD2
tff(fact_7461_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),P2: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P2),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,Xs),P2)),aa(nat,B,nth(B,Ys),P2)) ) ) ).

% list_all2_nthD
tff(fact_7462_list__all2__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(A,fun(B,$o)),Us: list(A),Vs: list(B)] :
      ( ( 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),list_all2(A,B,P),append(A,Xs,Us)),append(B,Ys,Vs))
      <=> ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Us),Vs) ) ) ) ).

% list_all2_append
tff(fact_7463_list__all2__append1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys: list(A),Zs: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),append(A,Xs,Ys)),Zs)
    <=> ? [Us3: list(B),Vs3: list(B)] :
          ( ( Zs = append(B,Us3,Vs3) )
          & ( aa(list(B),nat,size_size(list(B)),Us3) = aa(list(A),nat,size_size(list(A)),Xs) )
          & ( aa(list(B),nat,size_size(list(B)),Vs3) = aa(list(A),nat,size_size(list(A)),Ys) )
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Us3)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Ys),Vs3) ) ) ).

% list_all2_append1
tff(fact_7464_list__all2__append2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),Zs: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),append(B,Ys,Zs))
    <=> ? [Us3: list(A),Vs3: list(A)] :
          ( ( Xs = append(A,Us3,Vs3) )
          & ( aa(list(A),nat,size_size(list(A)),Us3) = aa(list(B),nat,size_size(list(B)),Ys) )
          & ( aa(list(A),nat,size_size(list(A)),Vs3) = aa(list(B),nat,size_size(list(B)),Zs) )
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Us3),Ys)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Vs3),Zs) ) ) ).

% list_all2_append2
tff(fact_7465_list__all2__lengthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys)
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% list_all2_lengthD
tff(fact_7466_pair__lessI1,axiom,
    ! [A3: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
     => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_7467_list__all2I,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),P: fun(A,fun(B,$o))] :
      ( ! [X4: product_prod(A,B)] :
          ( member(product_prod(A,B),X4,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,A3,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),P),X4) )
     => ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),A3),B2) ) ) ).

% list_all2I
tff(fact_7468_sum__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & monoid_add(A) )
     => ! [A4: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A4,zero_zero(A)),zero_zero(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),plus_plus(A)),plus_plus(B))
           => aa(fun(list(B),B),$o,aa(fun(list(A),A),fun(fun(list(B),B),$o),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A4),A4),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B)) ) ) ) ).

% sum_list_transfer
tff(fact_7469_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X: product_prod(A,B)] :
            ( member(product_prod(A,B),X,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),P),X) ) ) ) ).

% list_all2_iff
tff(fact_7470_prod__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_mult(B)
        & monoid_mult(A) )
     => ! [A4: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A4,one_one(A)),one_one(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),times_times(A)),times_times(B))
           => aa(fun(list(B),B),$o,aa(fun(list(A),A),fun(fun(list(B),B),$o),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A4),A4),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B)) ) ) ) ).

% prod_list_transfer
tff(fact_7471_pair__leqI2,axiom,
    ! [A3: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),S),Ta)
       => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_7472_prod__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A,Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(list(A),A,groups5270119922927024881d_list(A),Xs)) ) ).

% prod_list.Cons
tff(fact_7473_prod__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aa(list(A),A,groups5270119922927024881d_list(A),nil(A)) = one_one(A) ) ) ).

% prod_list.Nil
tff(fact_7474_prod__list_Oappend,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A),Ys: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),append(A,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups5270119922927024881d_list(A),Xs)),aa(list(A),A,groups5270119922927024881d_list(A),Ys)) ) ).

% prod_list.append
tff(fact_7475_prod__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),Xs) = foldr(A,A,times_times(A),Xs,one_one(A)) ) ).

% prod_list.eq_foldr
tff(fact_7476_pair__leqI1,axiom,
    ! [A3: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
     => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_7477_VEBT_Osimps_I7_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),F22: fun($o,fun($o,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : vEBT_rec_VEBT(A,F1,F22,vEBT_Node(X11,X12,X13,X14)) = aa(A,A,aa(vEBT_VEBT,fun(A,A),aa(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)),aa(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)))),F1,X11),X12),map(vEBT_VEBT,product_prod(vEBT_VEBT,A),aa(fun($o,fun($o,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_afd(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),F1),F22),X13)),X14),vEBT_rec_VEBT(A,F1,F22,X14)) ).

% VEBT.simps(7)
tff(fact_7478_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_afe(A,fun(A,$o)),aTP_Lamp_aff(A,fun(A,$o)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_7479_top_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ordering_top(A,ord_less_eq(A),ord_less(A),top_top(A)) ) ).

% top.ordering_top_axioms
tff(fact_7480_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_aj(nat,fun(nat,$o)),aTP_Lamp_ak(nat,fun(nat,$o)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_7481_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_7482_euclidean__size__times__nonunit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( ~ dvd_dvd(A,A3,one_one(A))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),B2)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_7483_euclidean__size__of__nat,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(A,nat,euclid6346220572633701492n_size(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = Nb ) ).

% euclidean_size_of_nat
tff(fact_7484_size__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ( aa(A,nat,euclid6346220572633701492n_size(A),zero_zero(A)) = zero_zero(nat) ) ) ).

% size_0
tff(fact_7485_euclidean__size__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( ( aa(A,nat,euclid6346220572633701492n_size(A),B2) = zero_zero(nat) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% euclidean_size_eq_0_iff
tff(fact_7486_euclidean__size__numeral,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [K: num] : aa(A,nat,euclid6346220572633701492n_size(A),aa(num,A,numeral_numeral(A),K)) = aa(num,nat,numeral_numeral(nat),K) ) ).

% euclidean_size_numeral
tff(fact_7487_euclidean__size__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) = one_one(nat) ) ) ).

% euclidean_size_1
tff(fact_7488_euclidean__size__greater__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
        <=> ( B2 != zero_zero(A) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_7489_euclidean__size__nat__def,axiom,
    euclid6346220572633701492n_size(nat) = id(nat) ).

% euclidean_size_nat_def
tff(fact_7490_euclidean__size__int__def,axiom,
    euclid6346220572633701492n_size(int) = aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)) ).

% euclidean_size_int_def
tff(fact_7491_euclidean__size__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] : aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2)) ) ).

% euclidean_size_mult
tff(fact_7492_euclidean__size__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) ) ) ) ).

% euclidean_size_unit
tff(fact_7493_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),B2) )
           => ( dvd_dvd(A,B2,A3)
             => dvd_dvd(A,A3,B2) ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_7494_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_7495_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_7496_unit__iff__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( dvd_dvd(A,A3,one_one(A))
        <=> ( ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) )
            & ( A3 != zero_zero(A) ) ) ) ) ).

% unit_iff_euclidean_size
tff(fact_7497_size__mult__mono_H,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3))) ) ) ).

% size_mult_mono'
tff(fact_7498_size__mult__mono,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ).

% size_mult_mono
tff(fact_7499_euclidean__size__times__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,nat,euclid6346220572633701492n_size(A),B2) ) ) ) ).

% euclidean_size_times_unit
tff(fact_7500_dvd__proper__imp__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( ~ dvd_dvd(A,B2,A3)
           => ( ( B2 != zero_zero(A) )
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2)) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_7501_dvd__imp__size__le,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( dvd_dvd(A,A3,B2)
         => ( ( B2 != zero_zero(A) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2)) ) ) ) ).

% dvd_imp_size_le
tff(fact_7502_mod__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),modulo_modulo(A,A3,B2))),aa(A,nat,euclid6346220572633701492n_size(A),B2)) ) ) ).

% mod_size_less
tff(fact_7503_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_7504_divmod__cases,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A3: A] :
          ( ( ( B2 != zero_zero(A) )
           => ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
             => ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2) ) ) )
         => ( ( ( B2 != zero_zero(A) )
             => ! [Q3: A,R3: A] :
                  ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R3)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
                   => ( ( R3 != zero_zero(A) )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q3 )
                       => ( ( modulo_modulo(A,A3,B2) = R3 )
                         => ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R3) ) ) ) ) ) ) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% divmod_cases
tff(fact_7505_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R) = A3 )
               => ( modulo_modulo(A,A3,B2) = R ) ) ) ) ) ) ).

% mod_eqI
tff(fact_7506_abs__division__segment,axiom,
    ! [K: int] : aa(int,int,abs_abs(int),euclid7384307370059645450egment(int,K)) = one_one(int) ).

% abs_division_segment
tff(fact_7507_division__segment__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( euclid7384307370059645450egment(A,one_one(A)) = one_one(A) ) ) ).

% division_segment_1
tff(fact_7508_division__segment__numeral,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [K: num] : euclid7384307370059645450egment(A,aa(num,A,numeral_numeral(A),K)) = one_one(A) ) ).

% division_segment_numeral
tff(fact_7509_division__segment__of__nat,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : euclid7384307370059645450egment(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = one_one(A) ) ).

% division_segment_of_nat
tff(fact_7510_division__segment__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A3)),aa(nat,A,semiring_1_of_nat(A),aa(A,nat,euclid6346220572633701492n_size(A),A3))) = A3 ) ).

% division_segment_euclidean_size
tff(fact_7511_division__segment__eq__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,B2: A] :
          ( ( euclid7384307370059645450egment(A,A3) = euclid7384307370059645450egment(A,B2) )
         => ( ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),B2) )
           => ( A3 = B2 ) ) ) ) ).

% division_segment_eq_iff
tff(fact_7512_division__segment__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( euclid7384307370059645450egment(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A3)),euclid7384307370059645450egment(A,B2)) ) ) ) ) ).

% division_segment_mult
tff(fact_7513_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( euclid7384307370059645450egment(int,K) = sgn_sgn(int,K) ) ) ).

% division_segment_eq_sgn
tff(fact_7514_division__segment__not__0,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A] : euclid7384307370059645450egment(A,A3) != zero_zero(A) ) ).

% division_segment_not_0
tff(fact_7515_division__segment__nat__def,axiom,
    ! [Nb: nat] : euclid7384307370059645450egment(nat,Nb) = one_one(nat) ).

% division_segment_nat_def
tff(fact_7516_is__unit__division__segment,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A] : dvd_dvd(A,euclid7384307370059645450egment(A,A3),one_one(A)) ) ).

% is_unit_division_segment
tff(fact_7517_division__segment__mod,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ~ dvd_dvd(A,B2,A3)
           => ( euclid7384307370059645450egment(A,modulo_modulo(A,A3,B2)) = euclid7384307370059645450egment(A,B2) ) ) ) ) ).

% division_segment_mod
tff(fact_7518_of__nat__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] : aa(nat,A,semiring_1_of_nat(A),aa(A,nat,euclid6346220572633701492n_size(A),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),euclid7384307370059645450egment(A,A3)) ) ).

% of_nat_euclidean_size
tff(fact_7519_division__segment__int__def,axiom,
    ! [K: int] :
      euclid7384307370059645450egment(int,K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ).

% division_segment_int_def
tff(fact_7520_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] :
          ( ( euclid7384307370059645450egment(A,A3) = euclid7384307370059645450egment(A,B2) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
          <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
              | ( B2 = zero_zero(A) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_7521_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R) = A3 )
               => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q2 ) ) ) ) ) ) ).

% div_eqI
tff(fact_7522_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R)),B2) = Q2 ) ) ) ) ) ).

% div_bounded
tff(fact_7523_find__dropWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : find(A,P,Xs) = case_list(option(A),A,none(A),aTP_Lamp_afg(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_7524_AboveS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : order_AboveS(A,R,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_afh(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R),A4)) ).

% AboveS_def
tff(fact_7525_wo__rel_Osuc__greater,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),field2(A,R))
       => ( ( order_AboveS(A,R,B4) != bot_bot(set(A)) )
         => ( member(A,B2,B4)
           => ( ( bNF_Wellorder_wo_suc(A,R,B4) != B2 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R,B4)),R) ) ) ) ) ) ).

% wo_rel.suc_greater
tff(fact_7526_wo__rel_Oequals__suc__AboveS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B4: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),field2(A,R))
       => ( member(A,A3,order_AboveS(A,R,B4))
         => ( ! [A10: A] :
                ( member(A,A10,order_AboveS(A,R,B4))
               => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A10),R) )
           => ( A3 = bNF_Wellorder_wo_suc(A,R,B4) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
tff(fact_7527_wo__rel_Osuc__least__AboveS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( member(A,A3,order_AboveS(A,R,B4))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R,B4)),A3),R) ) ) ).

% wo_rel.suc_least_AboveS
tff(fact_7528_wo__rel_Osuc__ofilter__in,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( order_ofilter(A,R,A4)
       => ( ( order_AboveS(A,R,A4) != bot_bot(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),B2),bNF_Wellorder_wo_suc(A,R,A4)),R)
           => ( ( B2 != bNF_Wellorder_wo_suc(A,R,A4) )
             => member(A,B2,A4) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
tff(fact_7529_less__eq__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Mb),Nb)
    <=> aa(extended_enat,$o,aa($o,fun(extended_enat,$o),aa(fun(nat,$o),fun($o,fun(extended_enat,$o)),extended_case_enat($o),aTP_Lamp_afi(extended_enat,fun(nat,$o),Mb)),$true),Nb) ) ).

% less_eq_enat_def
tff(fact_7530_less__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Mb),Nb)
    <=> aa(extended_enat,$o,aa($o,fun(extended_enat,$o),aa(fun(nat,$o),fun($o,fun(extended_enat,$o)),extended_case_enat($o),aTP_Lamp_afj(extended_enat,fun(nat,$o),Nb)),$false),Mb) ) ).

% less_enat_def
tff(fact_7531_ofilterIncl__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : bNF_We413866401316099525erIncl(A,R) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_afk(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R))) ).

% ofilterIncl_def
tff(fact_7532_sorted__wrt__iff__nth__Suc__transp,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( transp(A,P)
     => ( sorted_wrt(A,P,Xs)
      <=> ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))) ) ) ) ).

% sorted_wrt_iff_nth_Suc_transp
tff(fact_7533_transp__ge,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_afl(A,fun(A,$o))) ) ).

% transp_ge
tff(fact_7534_transp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less_eq(A)) ) ).

% transp_le
tff(fact_7535_transp__realrel,axiom,
    transp(fun(nat,rat),realrel) ).

% transp_realrel
tff(fact_7536_transp__gr,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_afm(A,fun(A,$o))) ) ).

% transp_gr
tff(fact_7537_transp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less(A)) ) ).

% transp_less
tff(fact_7538_bsqr__max2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A1: A,A22: A,B1: A,B22: A] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( member(product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),fun(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A))),product_Pair(product_prod(A,A),product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B1),B22)),bNF_Wellorder_bsqr(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),bNF_We1388413361240627857o_max2(A,R,A1,A22)),bNF_We1388413361240627857o_max2(A,R,B1,B22)),R) ) ) ).

% bsqr_max2
tff(fact_7539_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
        <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,aa(fun(product_prod(A,A),$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),ord_less(A))))) ) ) ).

% lexordp_conv_lexord
tff(fact_7540_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Xs: list(A),Y: A,Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
              & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_7541_well__order__on__domain,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_well_order_on(A,A4,R)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R)
       => ( member(A,A3,A4)
          & member(A,B2,A4) ) ) ) ).

% well_order_on_domain
tff(fact_7542_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Xs: list(A),Ys: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ).

% lexordp.Cons
tff(fact_7543_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Xs: list(A),Ys: list(A)] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
           => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_7544_lexordp__irreflexive,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),X4)
         => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Xs) ) ) ).

% lexordp_irreflexive
tff(fact_7545_lexordp__append__leftD,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A),Us: list(A),Vs: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),append(A,Xs,Us)),append(A,Xs,Vs))
         => ( ! [A5: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),A5)
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Us),Vs) ) ) ) ).

% lexordp_append_leftD
tff(fact_7546_natLeq__on__well__order__on,axiom,
    ! [Nb: nat] : order_well_order_on(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),Nb)),aa(fun(product_prod(nat,nat),$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,$o),aTP_Lamp_abz(nat,fun(nat,fun(nat,$o)),Nb)))) ).

% natLeq_on_well_order_on
tff(fact_7547_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
         => ( ( ( A1 = nil(A) )
             => ! [Y4: A,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
           => ( ! [X4: A] :
                  ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)
                 => ! [Y4: A] :
                      ( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4) ) )
             => ~ ! [X4: A,Y4: A,Xs2: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
                   => ! [Ys3: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                       => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
                         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X4)
                           => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_7548_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
        <=> ( ? [Y3: A,Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) ) )
            | ? [X: A,Y3: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) )
            | ? [X: A,Y3: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs3),Ys4) ) ) ) ) ).

% lexordp.simps
tff(fact_7549_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
         => ( ( ( Xs = nil(A) )
             => ! [Y4: A,Ys5: list(A)] : Ys != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys5) )
           => ( ! [X4: A] :
                  ( ? [Xs4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4)
                 => ! [Y4: A] :
                      ( ? [Ys5: list(A)] : Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys5)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4) ) )
             => ~ ! [X4: A,Xs4: list(A)] :
                    ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
                   => ! [Ys5: list(A)] :
                        ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys5) )
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs4),Ys5) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_7550_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A),P: fun(list(A),fun(list(A),$o))] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
         => ( ! [Y4: A,Ys3: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3))
           => ( ! [X4: A,Xs2: list(A),Y4: A,Ys3: list(A)] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
                 => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)) )
             => ( ! [X4: A,Xs2: list(A),Ys3: list(A)] :
                    ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3)
                   => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs2),Ys3)
                     => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3)) ) )
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs),Ys) ) ) ) ) ) ).

% lexordp_induct
tff(fact_7551_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
        <=> ( ? [X: A,Vs3: list(A)] : Ys = append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs3))
            | ? [Us3: list(A),A6: A,B5: A,Vs3: list(A),Ws3: list(A)] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A6),B5)
                & ( Xs = append(A,Us3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),Vs3)) )
                & ( Ys = append(A,Us3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B5),Ws3)) ) ) ) ) ) ).

% lexordp_iff
tff(fact_7552_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Us: list(A),Xs: list(A),Ys: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),append(A,Us,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs))),append(A,Us,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ) ).

% lexordp_append_left_rightI
tff(fact_7553_natLeq__on__Well__order,axiom,
    ! [Nb: nat] : order_well_order_on(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_abz(nat,fun(nat,fun(nat,$o)),Nb)))),aa(fun(product_prod(nat,nat),$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,$o),aTP_Lamp_abz(nat,fun(nat,fun(nat,$o)),Nb)))) ).

% natLeq_on_Well_order
tff(fact_7554_Linear__order__Well__order__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( order_well_order_on(A,field2(A,R),R)
      <=> ! [A11: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),field2(A,R))
           => ( ( A11 != bot_bot(set(A)) )
             => ? [X: A] :
                  ( member(A,X,A11)
                  & ! [Xa2: A] :
                      ( member(A,Xa2,A11)
                     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa2),R) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_7555_ofilter__subset__embedS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A4)
       => ( order_ofilter(A,R,B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
          <=> bNF_Wellorder_embedS(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_aec(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,B4,aTP_Lamp_aec(set(A),fun(A,set(A)),B4))),id(A)) ) ) ) ) ).

% ofilter_subset_embedS
tff(fact_7556_bsqr__ofilter,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),D5: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(product_prod(A,A),bNF_Wellorder_bsqr(A,R),D5)
       => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less(set(product_prod(A,A))),D5),product_Sigma(A,A,field2(A,R),aTP_Lamp_afn(set(product_prod(A,A)),fun(A,set(A)),R)))
         => ( ~ ? [A5: A] : field2(A,R) = order_under(A,R,A5)
           => ? [A8: set(A)] :
                ( order_ofilter(A,R,A8)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A8),field2(A,R))
                & aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),D5),product_Sigma(A,A,A8,aTP_Lamp_aec(set(A),fun(A,set(A)),A8))) ) ) ) ) ) ).

% bsqr_ofilter
tff(fact_7557_under__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_under(A,R,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_afo(set(product_prod(A,A)),fun(A,fun(A,$o)),R),A3)) ).

% under_def
tff(fact_7558_embedS__Field,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),F2: fun(A,B)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( bNF_Wellorder_embedS(A,B,R,R4,F2)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),image(A,B,F2,field2(A,R))),field2(B,R4)) ) ) ).

% embedS_Field
tff(fact_7559_ofilter__subset__embedS__iso,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A4)
       => ( order_ofilter(A,R,B4)
         => ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
            <=> bNF_Wellorder_embedS(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_aec(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,B4,aTP_Lamp_aec(set(A),fun(A,set(A)),B4))),id(A)) )
            & ( ( A4 = B4 )
            <=> bNF_Wellorder_iso(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_aec(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,B4,aTP_Lamp_aec(set(A),fun(A,set(A)),B4))),id(A)) ) ) ) ) ) ).

% ofilter_subset_embedS_iso
tff(fact_7560_iso__forward,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),F2: fun(A,B)] :
      ( 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)
     => ( bNF_Wellorder_iso(A,B,R,R4,F2)
       => member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)),R4) ) ) ).

% iso_forward
tff(fact_7561_iso__iff2,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),F2: fun(A,B)] :
      ( bNF_Wellorder_iso(A,B,R,R4,F2)
    <=> ( bij_betw(A,B,F2,field2(A,R),field2(B,R4))
        & ! [X: A] :
            ( member(A,X,field2(A,R))
           => ! [Xa2: A] :
                ( member(A,Xa2,field2(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),X),Xa2),R)
                <=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F2,X)),aa(A,B,F2,Xa2)),R4) ) ) ) ) ) ).

% iso_iff2
tff(fact_7562_ofilter__ordLess,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),field2(A,R))
        <=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_aec(set(A),fun(A,set(A)),A4)))),R),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ).

% ofilter_ordLess
tff(fact_7563_ofilter__subset__ordLess,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A4)
       => ( order_ofilter(A,R,B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B4)
          <=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_aec(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,B4,aTP_Lamp_aec(set(A),fun(A,set(A)),B4)))),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ) ).

% ofilter_subset_ordLess
tff(fact_7564_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_aad(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ) ).

% ord_class.lexordp_def
tff(fact_7565_embedS__iff,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),F2: fun(A,B)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( bNF_Wellorder_embed(A,B,R,R4,F2)
       => ( bNF_Wellorder_embedS(A,B,R,R4,F2)
        <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),image(A,B,F2,field2(A,R))),field2(B,R4)) ) ) ) ).

% embedS_iff
tff(fact_7566_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Nb: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2)) = complete_lattice_lfp(A,F2) ) ) ) ).

% lfp_funpow
tff(fact_7567_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F2),bot_bot(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_7568_lfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: fun(A,$o)] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( ! [S4: A] :
                ( aa(A,$o,P,S4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S4),complete_lattice_lfp(A,F2))
                 => aa(A,$o,P,aa(A,A,F2,S4)) ) )
           => ( ! [M7: set(A)] :
                  ( ! [X5: A] :
                      ( member(A,X5,M7)
                     => aa(A,$o,P,X5) )
                 => aa(A,$o,P,aa(set(A),A,complete_Sup_Sup(A),M7)) )
             => aa(A,$o,P,complete_lattice_lfp(A,F2)) ) ) ) ) ).

% lfp_ordinal_induct
tff(fact_7569_def__lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: A,F2: fun(A,A),P: A] :
          ( ( A4 = complete_lattice_lfp(A,F2) )
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),P))),P)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),P) ) ) ) ) ).

% def_lfp_induct
tff(fact_7570_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X6: fun($o,A),Y6: fun($o,A)] :
          ( aa(fun($o,A),$o,aa(fun($o,A),fun(fun($o,A),$o),ord_less_eq(fun($o,A)),X6),Y6)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,X6,$false)),aa($o,A,Y6,$false))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,X6,$true)),aa($o,A,Y6,$true)) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_7571_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,F2: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),complete_lattice_lfp(set(product_prod(A,B)),F2))
     => ( aa(fun(set(product_prod(A,B)),set(product_prod(A,B))),$o,order_mono(set(product_prod(A,B)),set(product_prod(A,B))),F2)
       => ( ! [A5: A,B3: B] :
              ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3),aa(set(product_prod(A,B)),set(product_prod(A,B)),F2,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),complete_lattice_lfp(set(product_prod(A,B)),F2)),aa(fun(product_prod(A,B),$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,$o),P)))))
             => aa(B,$o,aa(A,fun(B,$o),P,A5),B3) )
         => aa(B,$o,aa(A,fun(B,$o),P,A3),B2) ) ) ) ).

% lfp_induct2
tff(fact_7572_lfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),G: fun(A,A)] :
          ( ! [Z7: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,Z7)),aa(A,A,G,Z7))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),complete_lattice_lfp(A,G)) ) ) ).

% lfp_mono
tff(fact_7573_lfp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),A4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,A4)),A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),A4) ) ) ).

% lfp_lowerbound
tff(fact_7574_lfp__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),A4: A] :
          ( ! [U3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,U3)),U3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),U3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),complete_lattice_lfp(A,F2)) ) ) ).

% lfp_greatest
tff(fact_7575_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,fun(A,A))] :
          ( ! [X4: A,Y4: A,W2: A,Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X4),W2)),aa(A,A,aa(A,fun(A,A),F2,Y4),Z3)) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_afp(fun(A,fun(A,A)),fun(A,A),F2)) = complete_lattice_lfp(A,aTP_Lamp_afq(fun(A,fun(A,A)),fun(A,A),F2)) ) ) ) ).

% lfp_lfp
tff(fact_7576_lfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),Xb: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F4)
         => ( ( aa(A,A,F4,Xb) = Xb )
           => ( ! [Z3: A] :
                  ( ( aa(A,A,F4,Z3) = Z3 )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z3) )
             => ( complete_lattice_lfp(A,F4) = Xb ) ) ) ) ) ).

% lfp_eqI
tff(fact_7577_lfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A)] : complete_lattice_lfp(A,F2) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_afr(fun(A,A),fun(A,$o),F2))) ) ).

% lfp_def
tff(fact_7578_lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F2)),P))),P)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),P) ) ) ) ).

% lfp_induct
tff(fact_7579_lfp__transfer__bounded,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [P: fun(A,$o),F2: fun(A,A),Alpha: fun(A,B),G: fun(B,B)] :
          ( aa(A,$o,P,bot_bot(A))
         => ( ! [X4: A] :
                ( aa(A,$o,P,X4)
               => aa(A,$o,P,aa(A,A,F2,X4)) )
           => ( ! [M7: fun(nat,A)] :
                  ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M7,I3))
                 => aa(A,$o,P,aa(set(A),A,complete_Sup_Sup(A),image(nat,A,M7,top_top(set(nat))))) )
             => ( ! [M7: fun(nat,A)] :
                    ( aa(fun(nat,A),$o,order_mono(nat,A),M7)
                   => ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M7,I3))
                     => ( aa(A,B,Alpha,aa(set(A),A,complete_Sup_Sup(A),image(nat,A,M7,top_top(set(nat))))) = aa(set(B),B,complete_Sup_Sup(B),image(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_afs(fun(A,B),fun(fun(nat,A),fun(nat,B)),Alpha),M7),top_top(set(nat)))) ) ) )
               => ( order_sup_continuous(A,A,F2)
                 => ( order_sup_continuous(B,B,G)
                   => ( ! [X4: A] :
                          ( aa(A,$o,P,X4)
                         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),complete_lattice_lfp(A,F2))
                           => ( aa(A,B,Alpha,aa(A,A,F2,X4)) = aa(B,B,G,aa(A,B,Alpha,X4)) ) ) )
                     => ( ! [X4: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G,X4))
                       => ( aa(A,B,Alpha,complete_lattice_lfp(A,F2)) = complete_lattice_lfp(B,G) ) ) ) ) ) ) ) ) ) ) ).

% lfp_transfer_bounded
tff(fact_7580_bounded__bilinear__def,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
        <=> ( ! [A6: A,A14: A,B5: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),A14)),B5) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A6),B5)),aa(B,C,aa(A,fun(B,C),Prod,A14),B5))
            & ! [A6: A,B5: B,B12: B] : aa(B,C,aa(A,fun(B,C),Prod,A6),aa(B,B,aa(B,fun(B,B),plus_plus(B),B5),B12)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A6),B5)),aa(B,C,aa(A,fun(B,C),Prod,A6),B12))
            & ! [R5: real,A6: A,B5: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,real_V8093663219630862766scaleR(A,R5),A6)),B5) = aa(C,C,real_V8093663219630862766scaleR(C,R5),aa(B,C,aa(A,fun(B,C),Prod,A6),B5))
            & ! [A6: A,R5: real,B5: B] : aa(B,C,aa(A,fun(B,C),Prod,A6),aa(B,B,real_V8093663219630862766scaleR(B,R5),B5)) = aa(C,C,real_V8093663219630862766scaleR(C,R5),aa(B,C,aa(A,fun(B,C),Prod,A6),B5))
            & ? [K6: real] :
              ! [A6: A,B5: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A6),B5))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A6)),real_V7770717601297561774m_norm(B,B5))),K6)) ) ) ) ).

% bounded_bilinear_def
tff(fact_7581_bounded__bilinear__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => real_V2442710119149674383linear(A,A,A,times_times(A)) ) ).

% bounded_bilinear_mult
tff(fact_7582_bounded__bilinear_Oadd__right,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),A3: A,B2: B,B6: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),B6)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A3),B2)),aa(B,C,aa(A,fun(B,C),Prod,A3),B6)) ) ) ) ).

% bounded_bilinear.add_right
tff(fact_7583_bounded__bilinear_Oadd__left,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),A3: A,A7: A,B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A7)),B2) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A3),B2)),aa(B,C,aa(A,fun(B,C),Prod,A7),B2)) ) ) ) ).

% bounded_bilinear.add_left
tff(fact_7584_bounded__bilinear_Oprod__diff__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),Xb: A,Y: B,A3: A,B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(B,C,aa(A,fun(B,C),Prod,Xb),Y)),aa(B,C,aa(A,fun(B,C),Prod,A3),B2)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A3)),aa(B,B,aa(B,fun(B,B),minus_minus(B),Y),B2))),aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A3)),B2))),aa(B,C,aa(A,fun(B,C),Prod,A3),aa(B,B,aa(B,fun(B,B),minus_minus(B),Y),B2))) ) ) ) ).

% bounded_bilinear.prod_diff_prod
tff(fact_7585_bounded__bilinear_OFDERIV,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,A),F6: fun(D,A),Xb: D,S: set(D),G: fun(D,B),G5: fun(D,B)] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( has_derivative(D,A,F2,F6,topolo174197925503356063within(D,Xb,S))
           => ( has_derivative(D,B,G,G5,topolo174197925503356063within(D,Xb,S))
             => has_derivative(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_aft(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Prod),F2),G),aa(fun(D,B),fun(D,C),aa(fun(D,B),fun(fun(D,B),fun(D,C)),aa(D,fun(fun(D,B),fun(fun(D,B),fun(D,C))),aa(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C)))),aa(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C))))),aTP_Lamp_afu(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C)))))),Prod),F2),F6),Xb),G),G5),topolo174197925503356063within(D,Xb,S)) ) ) ) ) ).

% bounded_bilinear.FDERIV
tff(fact_7586_lfp__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [Alpha: fun(A,B),F2: fun(A,A),G: fun(B,B)] :
          ( order_sup_continuous(A,B,Alpha)
         => ( order_sup_continuous(A,A,F2)
           => ( order_sup_continuous(B,B,G)
             => ( ! [X4: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G,X4))
               => ( ! [X4: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),complete_lattice_lfp(A,F2))
                     => ( aa(A,B,Alpha,aa(A,A,F2,X4)) = aa(B,B,G,aa(A,B,Alpha,X4)) ) )
                 => ( aa(A,B,Alpha,complete_lattice_lfp(A,F2)) = complete_lattice_lfp(B,G) ) ) ) ) ) ) ) ).

% lfp_transfer
tff(fact_7587_bounded__bilinear_Opos__bounded,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [A9: A,B10: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A9),B10))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A9)),real_V7770717601297561774m_norm(B,B10))),K8)) ) ) ) ).

% bounded_bilinear.pos_bounded
tff(fact_7588_bounded__bilinear_Ointro,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( ! [A5: A,A10: A,B3: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),A10)),B3) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A5),B3)),aa(B,C,aa(A,fun(B,C),Prod,A10),B3))
         => ( ! [A5: A,B3: B,B8: B] : aa(B,C,aa(A,fun(B,C),Prod,A5),aa(B,B,aa(B,fun(B,B),plus_plus(B),B3),B8)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A5),B3)),aa(B,C,aa(A,fun(B,C),Prod,A5),B8))
           => ( ! [R3: real,A5: A,B3: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,real_V8093663219630862766scaleR(A,R3),A5)),B3) = aa(C,C,real_V8093663219630862766scaleR(C,R3),aa(B,C,aa(A,fun(B,C),Prod,A5),B3))
             => ( ! [A5: A,R3: real,B3: B] : aa(B,C,aa(A,fun(B,C),Prod,A5),aa(B,B,real_V8093663219630862766scaleR(B,R3),B3)) = aa(C,C,real_V8093663219630862766scaleR(C,R3),aa(B,C,aa(A,fun(B,C),Prod,A5),B3))
               => ( ? [K9: real] :
                    ! [A5: A,B3: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A5),B3))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A5)),real_V7770717601297561774m_norm(B,B3))),K9))
                 => real_V2442710119149674383linear(A,B,C,Prod) ) ) ) ) ) ) ).

% bounded_bilinear.intro
tff(fact_7589_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( condit941137186595557371_above(A,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ) ) ).

% mono_cSup
tff(fact_7590_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A4: fun(C,A),I5: set(C)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( condit941137186595557371_above(A,image(C,A,A4,I5))
           => ( ( I5 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afv(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),image(C,A,A4,I5)))) ) ) ) ) ).

% mono_cSUP
tff(fact_7591_bdd__above_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A),M6: A] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),M6) )
         => condit941137186595557371_above(A,A4) ) ) ).

% bdd_above.I
tff(fact_7592_bdd__above_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A)] :
          ( condit941137186595557371_above(A,A4)
        <=> ? [M8: A] :
            ! [X: A] :
              ( member(A,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),M8) ) ) ) ).

% bdd_above.unfold
tff(fact_7593_bdd__above_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A)] :
          ( condit941137186595557371_above(A,A4)
         => ~ ! [M7: A] :
                ~ ! [X5: A] :
                    ( member(A,X5,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),M7) ) ) ) ).

% bdd_above.E
tff(fact_7594_cSup__upper,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X6: set(A)] :
          ( member(A,Xb,X6)
         => ( condit941137186595557371_above(A,X6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),X6)) ) ) ) ).

% cSup_upper
tff(fact_7595_cSup__upper2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X6: set(A),Y: A] :
          ( member(A,Xb,X6)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
           => ( condit941137186595557371_above(A,X6)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6)) ) ) ) ) ).

% cSup_upper2
tff(fact_7596_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B4: set(A),A4: set(A)] :
          ( ( B4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A4)
           => ( ! [B3: A] :
                  ( member(A,B3,B4)
                 => ? [X5: A] :
                      ( member(A,X5,A4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),X5) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B4)),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ) ).

% cSup_mono
tff(fact_7597_cSup__le__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S3: set(A),A3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S3)),A3)
            <=> ! [X: A] :
                  ( member(A,X,S3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_7598_cSUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Xb: A,A4: set(A),F2: fun(A,B)] :
          ( member(A,Xb,A4)
         => ( condit941137186595557371_above(B,image(A,B,F2,A4))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))) ) ) ) ).

% cSUP_upper
tff(fact_7599_cSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Xb: B,U: A] :
          ( condit941137186595557371_above(A,image(B,A,F2,A4))
         => ( member(B,Xb,A4)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F2,Xb))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))) ) ) ) ) ).

% cSUP_upper2
tff(fact_7600_bdd__above_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A4: set(A),F2: fun(A,B),M6: B] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),M6) )
         => condit941137186595557371_above(B,image(A,B,F2,A4)) ) ) ).

% bdd_above.I2
tff(fact_7601_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X6)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6))
            <=> ? [X: A] :
                  ( member(A,X,X6)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_7602_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Y: A,I: B] :
          ( condit941137186595557371_above(A,image(B,A,F2,A4))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))),Y)
           => ( member(B,I,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,I)),Y) ) ) ) ) ).

% cSUP_lessD
tff(fact_7603_cSUP__mono,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),G: fun(C,B),B4: set(C),F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,image(C,B,G,B4))
           => ( ! [N: A] :
                  ( member(A,N,A4)
                 => ? [X5: C] :
                      ( member(C,X5,B4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,N)),aa(C,B,G,X5)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))),aa(set(B),B,complete_Sup_Sup(B),image(C,B,G,B4))) ) ) ) ) ).

% cSUP_mono
tff(fact_7604_cSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),U: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,image(A,B,F2,A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))),U)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),U) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_7605_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,B4)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B4)) ) ) ) ) ).

% cSup_subset_mono
tff(fact_7606_less__cSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A4: set(A),F2: fun(A,B),A3: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,image(A,B,F2,A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4)))
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(A,B,F2,X)) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_7607_cSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),G: fun(A,B),B4: set(A),F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,image(A,B,G,B4))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
             => ( ! [X4: A] :
                    ( member(A,X4,A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A4))),aa(set(B),B,complete_Sup_Sup(B),image(A,B,G,B4))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_7608_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B4: set(A)] :
          ( condit941137186595557371_above(A,A4)
         => ( condit941137186595557371_above(A,B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_7609_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S3: set(A)] :
          ( ( S3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S3)
           => ( aa(set(A),A,complete_Sup_Sup(A),S3) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_afw(set(A),fun(A,$o),S3))) ) ) ) ) ).

% cSup_cInf
tff(fact_7610_prod__decode__triangle__add,axiom,
    ! [K: nat,Mb: nat] : nat_prod_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),Mb)) = nat_prod_decode_aux(K,Mb) ).

% prod_decode_triangle_add
tff(fact_7611_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F2: fun(A,B),A4: fun(C,A),I5: set(C)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( condit1013018076250108175_below(A,image(C,A,A4,I5))
           => ( ( I5 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),image(C,A,A4,I5)))),aa(set(B),B,complete_Inf_Inf(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afv(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4),I5))) ) ) ) ) ).

% mono_cINF
tff(fact_7612_bdd__belowI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A),Mb: A] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X4) )
         => condit1013018076250108175_below(A,A4) ) ) ).

% bdd_belowI
tff(fact_7613_bdd__below_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A),M6: A] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M6),X4) )
         => condit1013018076250108175_below(A,A4) ) ) ).

% bdd_below.I
tff(fact_7614_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X6)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y)
            <=> ? [X: A] :
                  ( member(A,X,X6)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_7615_cINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Xb: B] :
          ( condit1013018076250108175_below(A,image(B,A,F2,A4))
         => ( member(B,Xb,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))),aa(B,A,F2,Xb)) ) ) ) ).

% cINF_lower
tff(fact_7616_cINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Xb: B,U: A] :
          ( condit1013018076250108175_below(A,image(B,A,F2,A4))
         => ( member(B,Xb,A4)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,Xb)),U)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))),U) ) ) ) ) ).

% cINF_lower2
tff(fact_7617_bdd__belowI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A4: set(A),Mb: B,F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(A,B,F2,X4)) )
         => condit1013018076250108175_below(B,image(A,B,F2,A4)) ) ) ).

% bdd_belowI2
tff(fact_7618_bdd__below_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A4: set(A),M6: B,F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M6),aa(A,B,F2,X4)) )
         => condit1013018076250108175_below(B,image(A,B,F2,A4)) ) ) ).

% bdd_below.I2
tff(fact_7619_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B4: set(A),A4: set(A)] :
          ( ( B4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A4)
           => ( ! [B3: A] :
                  ( member(A,B3,B4)
                 => ? [X5: A] :
                      ( member(A,X5,A4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),B3) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B4)) ) ) ) ) ).

% cInf_mono
tff(fact_7620_le__cInf__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S3: set(A),A3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,complete_Inf_Inf(A),S3))
            <=> ! [X: A] :
                  ( member(A,X,S3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_7621_cInf__lower2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X6: set(A),Y: A] :
          ( member(A,Xb,X6)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( condit1013018076250108175_below(A,X6)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y) ) ) ) ) ).

% cInf_lower2
tff(fact_7622_cInf__lower,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X6: set(A)] :
          ( member(A,Xb,X6)
         => ( condit1013018076250108175_below(A,X6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Xb) ) ) ) ).

% cInf_lower
tff(fact_7623_bdd__below_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A)] :
          ( condit1013018076250108175_below(A,A4)
         => ~ ! [M7: A] :
                ~ ! [X5: A] :
                    ( member(A,X5,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M7),X5) ) ) ) ).

% bdd_below.E
tff(fact_7624_bdd__below_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A)] :
          ( condit1013018076250108175_below(A,A4)
        <=> ? [M8: A] :
            ! [X: A] :
              ( member(A,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M8),X) ) ) ) ).

% bdd_below.unfold
tff(fact_7625_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Y: A,I: B] :
          ( condit1013018076250108175_below(A,image(B,A,F2,A4))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4)))
           => ( member(B,I,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,I)) ) ) ) ) ).

% less_cINF_D
tff(fact_7626_cINF__mono,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [B4: set(A),F2: fun(C,B),A4: set(C),G: fun(A,B)] :
          ( ( B4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,image(C,B,F2,A4))
           => ( ! [M2: A] :
                  ( member(A,M2,B4)
                 => ? [X5: C] :
                      ( member(C,X5,A4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F2,X5)),aa(A,B,G,M2)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(C,B,F2,A4))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,G,B4))) ) ) ) ) ).

% cINF_mono
tff(fact_7627_le__cINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),U: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,image(A,B,F2,A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4)))
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,X)) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_7628_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,B4)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B4)),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ) ) ) ).

% cInf_superset_mono
tff(fact_7629_cINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A4: set(A),F2: fun(A,B),A3: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,image(A,B,F2,A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))),A3)
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),A3) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_7630_cINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),G: fun(A,B),B4: set(A),F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,image(A,B,G,B4))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4)
             => ( ! [X4: A] :
                    ( member(A,X4,B4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,G,X4)),aa(A,B,F2,X4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,G,B4))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_7631_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B4: set(A)] :
          ( condit1013018076250108175_below(A,A4)
         => ( condit1013018076250108175_below(A,B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B4))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_7632_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A4)
           => ( condit1013018076250108175_below(A,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ) ).

% cInf_le_cSup
tff(fact_7633_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S3: set(A)] :
          ( ( S3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S3)
           => ( aa(set(A),A,complete_Inf_Inf(A),S3) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_afx(set(A),fun(A,$o),S3))) ) ) ) ) ).

% cInf_cSup
tff(fact_7634_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( condit1013018076250108175_below(A,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A4))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A4))) ) ) ) ) ).

% mono_cInf
tff(fact_7635_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun(nat,$o)] :
      ( accp(nat,nat_list_decode_rel,A0)
     => ( ( accp(nat,nat_list_decode_rel,zero_zero(nat))
         => aa(nat,$o,P,zero_zero(nat)) )
       => ( ! [N: nat] :
              ( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N))
             => ( ! [X5: nat,Y5: nat] :
                    ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X5),Y5) = nat_prod_decode(N) )
                   => aa(nat,$o,P,Y5) )
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,A0) ) ) ) ).

% list_decode.pinduct
tff(fact_7636_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_kw(A,fun(A,$o)),aTP_Lamp_afy(A,fun(A,$o))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_7637_Inf__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => lattic4895041142388067077er_set(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_7638_Min_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).

% Min.semilattice_order_set_axioms
tff(fact_7639_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_afz(A,fun(A,$o)),aTP_Lamp_aga(A,fun(A,$o))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_7640_Gcd__fin__def,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A) = bounde2362111253966948842tice_F(A,gcd_gcd(A),zero_zero(A),one_one(A)) ) ) ).

% Gcd_fin_def
tff(fact_7641_flip__pred,axiom,
    ! [A: $tType,B: $tType,A4: set(product_prod(A,B)),R2: fun(B,fun(A,$o))] :
      ( 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))),A4),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),conversep(B,A,R2))))
     => 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))),image(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_aam(A,fun(B,product_prod(B,A)))),A4)),aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),R2))) ) ).

% flip_pred
tff(fact_7642_compute__powr__real,axiom,
    ! [B2: real,I: real] :
      powr_real(B2,I) = $ite(
        aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),zero_zero(real)),
        abort(real,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$false,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,zero_zero(literal)))))))))))))))))))))))))))))))),aa(real,fun(product_unit,real),aTP_Lamp_agb(real,fun(real,fun(product_unit,real)),B2),I)),
        $ite(
          aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,I)) = I,
          $ite(aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),I),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(int,nat,nat2,archim6421214686448440834_floor(real,I))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(int,nat,nat2,archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),I)))))),
          abort(real,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$true,$false,$true,$false,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$true,$true,$false,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$true,$true,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,zero_zero(literal)))))))))))))))))))))))))))))))))))),aa(real,fun(product_unit,real),aTP_Lamp_agb(real,fun(real,fun(product_unit,real)),B2),I)) ) ) ).

% compute_powr_real
tff(fact_7643_map__comp__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,option(B)),G: fun(A,option(C)),X5: A] : map_comp(C,B,A,F2,G,X5) = aa(option(C),option(B),aa(fun(C,option(B)),fun(option(C),option(B)),aa(option(B),fun(fun(C,option(B)),fun(option(C),option(B))),case_option(option(B),C),none(B)),F2),aa(A,option(C),G,X5)) ).

% map_comp_def
tff(fact_7644_map__comp__simps_I2_J,axiom,
    ! [B: $tType,C: $tType,A: $tType,M22: fun(B,option(A)),K: B,K7: A,M1: fun(A,option(C))] :
      ( ( aa(B,option(A),M22,K) = aa(A,option(A),some(A),K7) )
     => ( map_comp(A,C,B,M1,M22,K) = aa(A,option(C),M1,K7) ) ) ).

% map_comp_simps(2)
tff(fact_7645_map__comp__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,C: $tType,M22: fun(B,option(A)),K: B,M1: fun(A,option(C))] :
      ( ( aa(B,option(A),M22,K) = none(A) )
     => ( map_comp(A,C,B,M1,M22,K) = none(C) ) ) ).

% map_comp_simps(1)
tff(fact_7646_map__comp__empty_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Mb: fun(A,option(C)),X5: A] : map_comp(C,B,A,aTP_Lamp_agc(C,option(B)),Mb,X5) = none(B) ).

% map_comp_empty(2)
tff(fact_7647_map__comp__empty_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Mb: fun(C,option(B)),X5: A] : map_comp(C,B,A,Mb,aTP_Lamp_aas(A,option(C)),X5) = none(B) ).

% map_comp_empty(1)
tff(fact_7648_map__comp__Some__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,M1: fun(B,option(A)),M22: fun(C,option(B)),K: C,V: A] :
      ( ( map_comp(B,A,C,M1,M22,K) = aa(A,option(A),some(A),V) )
    <=> ? [K10: B] :
          ( ( aa(C,option(B),M22,K) = aa(B,option(B),some(B),K10) )
          & ( aa(B,option(A),M1,K10) = aa(A,option(A),some(A),V) ) ) ) ).

% map_comp_Some_iff
tff(fact_7649_map__comp__None__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,M1: fun(B,option(A)),M22: fun(C,option(B)),K: C] :
      ( ( map_comp(B,A,C,M1,M22,K) = none(A) )
    <=> ( ( aa(C,option(B),M22,K) = none(B) )
        | ? [K10: B] :
            ( ( aa(C,option(B),M22,K) = aa(B,option(B),some(B),K10) )
            & ( aa(B,option(A),M1,K10) = none(A) ) ) ) ) ).

% map_comp_None_iff
tff(fact_7650_gen__length__def,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),nat,gen_length(A,Nb),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)) ).

% gen_length_def
tff(fact_7651_relImage__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,B)),F2: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R2,F2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_agd(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),R2),F2)) ).

% relImage_def
tff(fact_7652_length__code,axiom,
    ! [A: $tType] : size_size(list(A)) = gen_length(A,zero_zero(nat)) ).

% length_code
tff(fact_7653_finite__subset__Union__chain,axiom,
    ! [A: $tType,A4: set(A),B13: set(set(A)),A15: set(set(A))] :
      ( finite_finite(A,A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B13))
       => ( ( B13 != bot_bot(set(set(A))) )
         => ( aa(set(set(A)),$o,pred_chain(set(A),A15,ord_less(set(A))),B13)
           => ~ ! [B9: set(A)] :
                  ( member(set(A),B9,B13)
                 => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B9) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_7654_irrefl__tranclI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xb: A] :
      ( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R)),transitive_rtrancl(A,R)) = bot_bot(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),transitive_trancl(A,R)) ) ).

% irrefl_tranclI
tff(fact_7655_converse__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),converse(B,A,R))
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3),R) ) ).

% converse_iff
tff(fact_7656_conversep__converse__eq,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),X5: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),conversep(B,A,aTP_Lamp_age(set(product_prod(B,A)),fun(B,fun(A,$o)),R)),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa),converse(B,A,R)) ) ).

% conversep_converse_eq
tff(fact_7657_subset__Zorn,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ! [C7: set(set(A))] :
          ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C7)
         => ? [X5: set(A)] :
              ( member(set(A),X5,A4)
              & ! [Xa3: set(A)] :
                  ( member(set(A),Xa3,C7)
                 => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa3),X5) ) ) )
     => ? [X4: set(A)] :
          ( member(set(A),X4,A4)
          & ! [Xa: set(A)] :
              ( member(set(A),Xa,A4)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),Xa)
               => ( Xa = X4 ) ) ) ) ) ).

% subset_Zorn
tff(fact_7658_subset__chain__insert,axiom,
    ! [A: $tType,A15: set(set(A)),B4: set(A),B13: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A15,ord_less(set(A))),aa(set(set(A)),set(set(A)),insert(set(A),B4),B13))
    <=> ( member(set(A),B4,A15)
        & ! [X: set(A)] :
            ( member(set(A),X,B13)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),B4)
              | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),X) ) )
        & aa(set(set(A)),$o,pred_chain(set(A),A15,ord_less(set(A))),B13) ) ) ).

% subset_chain_insert
tff(fact_7659_subset_Ochain__def,axiom,
    ! [A: $tType,A4: set(set(A)),C5: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C5)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C5),A4)
        & ! [X: set(A)] :
            ( member(set(A),X,C5)
           => ! [Xa2: set(A)] :
                ( member(set(A),Xa2,C5)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X),Xa2)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Xa2),X) ) ) ) ) ) ).

% subset.chain_def
tff(fact_7660_subset_OchainI,axiom,
    ! [A: $tType,C5: set(set(A)),A4: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C5),A4)
     => ( ! [X4: set(A),Y4: set(A)] :
            ( member(set(A),X4,C5)
           => ( member(set(A),Y4,C5)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X4),Y4)
                | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Y4),X4) ) ) )
       => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C5) ) ) ).

% subset.chainI
tff(fact_7661_subset__Zorn_H,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ! [C7: set(set(A))] :
          ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C7)
         => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C7),A4) )
     => ? [X4: set(A)] :
          ( member(set(A),X4,A4)
          & ! [Xa: set(A)] :
              ( member(set(A),Xa,A4)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),Xa)
               => ( Xa = X4 ) ) ) ) ) ).

% subset_Zorn'
tff(fact_7662_subset__chain__def,axiom,
    ! [A: $tType,A15: set(set(A)),C8: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A15,ord_less(set(A))),C8)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C8),A15)
        & ! [X: set(A)] :
            ( member(set(A),X,C8)
           => ! [Xa2: set(A)] :
                ( member(set(A),Xa2,C8)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),Xa2)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa2),X) ) ) ) ) ) ).

% subset_chain_def
tff(fact_7663_subset_Ochain__empty,axiom,
    ! [A: $tType,A4: set(set(A))] : aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),bot_bot(set(set(A)))) ).

% subset.chain_empty
tff(fact_7664_subset_Ochain__total,axiom,
    ! [A: $tType,A4: set(set(A)),C5: set(set(A)),Xb: set(A),Y: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C5)
     => ( member(set(A),Xb,C5)
       => ( member(set(A),Y,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Xb),Y)
            | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Y),Xb) ) ) ) ) ).

% subset.chain_total
tff(fact_7665_subset_Ochain__extend,axiom,
    ! [A: $tType,A4: set(set(A)),C5: set(set(A)),Z: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C5)
     => ( member(set(A),Z,A4)
       => ( ! [X4: set(A)] :
              ( member(set(A),X4,C5)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X4),Z) )
         => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),insert(set(A),Z),bot_bot(set(set(A))))),C5)) ) ) ) ).

% subset.chain_extend
tff(fact_7666_trancl__converseI,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),converse(A,A,transitive_trancl(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),Xb),Y),transitive_trancl(A,converse(A,A,R))) ) ).

% trancl_converseI
tff(fact_7667_trancl__converseD,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,converse(A,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),Xb),Y),converse(A,A,transitive_trancl(A,R))) ) ).

% trancl_converseD
tff(fact_7668_rtrancl__converseI,axiom,
    ! [A: $tType,Y: A,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),Y),Xb),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),Xb),Y),transitive_rtrancl(A,converse(A,A,R))) ) ).

% rtrancl_converseI
tff(fact_7669_rtrancl__converseD,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,converse(A,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),Y),Xb),transitive_rtrancl(A,R)) ) ).

% rtrancl_converseD
tff(fact_7670_converse__unfold,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A))] : converse(B,A,R) = aa(fun(product_prod(A,B),$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,$o),aTP_Lamp_agf(set(product_prod(B,A)),fun(A,fun(B,$o)),R))) ).

% converse_unfold
tff(fact_7671_converse_Ocases,axiom,
    ! [A: $tType,B: $tType,A1: A,A22: B,R: set(product_prod(B,A))] :
      ( 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),converse(B,A,R))
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A22),A1),R) ) ).

% converse.cases
tff(fact_7672_converse_Osimps,axiom,
    ! [A: $tType,B: $tType,A1: A,A22: B,R: set(product_prod(B,A))] :
      ( 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),converse(B,A,R))
    <=> ? [A6: B,B5: A] :
          ( ( A1 = B5 )
          & ( A22 = A6 )
          & member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A6),B5),R) ) ) ).

% converse.simps
tff(fact_7673_converseD,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),converse(B,A,R))
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3),R) ) ).

% converseD
tff(fact_7674_converseE,axiom,
    ! [A: $tType,B: $tType,Yx: product_prod(A,B),R: set(product_prod(B,A))] :
      ( member(product_prod(A,B),Yx,converse(B,A,R))
     => ~ ! [X4: B,Y4: A] :
            ( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y4),X4) )
           => ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),Y4),R) ) ) ).

% converseE
tff(fact_7675_converseI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(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),A3),B2),R)
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3),converse(A,B,R)) ) ).

% converseI
tff(fact_7676_converse__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : converse(A,B,X5) = aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),X5)))) ).

% converse_def
tff(fact_7677_subset__Zorn__nonempty,axiom,
    ! [A: $tType,A15: set(set(A))] :
      ( ( A15 != bot_bot(set(set(A))) )
     => ( ! [C9: set(set(A))] :
            ( ( C9 != bot_bot(set(set(A))) )
           => ( aa(set(set(A)),$o,pred_chain(set(A),A15,ord_less(set(A))),C9)
             => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C9),A15) ) )
       => ? [X4: set(A)] :
            ( member(set(A),X4,A15)
            & ! [Xa: set(A)] :
                ( member(set(A),Xa,A15)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),Xa)
                 => ( Xa = X4 ) ) ) ) ) ) ).

% subset_Zorn_nonempty
tff(fact_7678_Union__in__chain,axiom,
    ! [A: $tType,B13: set(set(A)),A15: set(set(A))] :
      ( finite_finite(set(A),B13)
     => ( ( B13 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A15,ord_less(set(A))),B13)
         => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B13),B13) ) ) ) ).

% Union_in_chain
tff(fact_7679_Inter__in__chain,axiom,
    ! [A: $tType,B13: set(set(A)),A15: set(set(A))] :
      ( finite_finite(set(A),B13)
     => ( ( B13 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A15,ord_less(set(A))),B13)
         => member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B13),B13) ) ) ) ).

% Inter_in_chain
tff(fact_7680_Chains__subset_H,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,$o)),R)))),chains(A,R)) ) ).

% Chains_subset'
tff(fact_7681_Chains__alt__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R)
     => ( chains(A,R) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,$o)),R))) ) ) ).

% Chains_alt_def
tff(fact_7682_Chains__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : chains(A,R) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_agg(set(product_prod(A,A)),fun(set(A),$o),R)) ).

% Chains_def
tff(fact_7683_Chains__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),chains(A,R)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,$o)),R)))) ).

% Chains_subset
tff(fact_7684_chain__subset__alt__def,axiom,
    ! [A: $tType,C5: set(set(A))] :
      ( chain_subset(A,C5)
    <=> aa(set(set(A)),$o,pred_chain(set(A),top_top(set(set(A))),ord_less(set(A))),C5) ) ).

% chain_subset_alt_def
tff(fact_7685_chains__alt__def,axiom,
    ! [A: $tType,A4: set(set(A))] : chains2(A,A4) = aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),pred_chain(set(A),A4,ord_less(set(A)))) ).

% chains_alt_def
tff(fact_7686_prod__set__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,Xb: B,Y: A] : basic_snds(B,A,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y)) = aa(set(A),set(A),insert(A,Y),bot_bot(set(A))) ).

% prod_set_simps(2)
tff(fact_7687_prod__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B] : basic_fsts(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y)) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) ).

% prod_set_simps(1)
tff(fact_7688_numeral__le__enat__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( 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)),extended_enat2(Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Mb)),Nb) ) ).

% numeral_le_enat_iff
tff(fact_7689_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_agh(A,$o)) ) ) ).

% Nats_altdef2
tff(fact_7690_enat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( extended_enat2(Nat) = extended_enat2(Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% enat.inject
tff(fact_7691_enat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: fun(nat,A),F22: A,Nat: nat] : aa(extended_enat,A,aa(A,fun(extended_enat,A),aa(fun(nat,A),fun(A,fun(extended_enat,A)),extended_case_enat(A),F1),F22),extended_enat2(Nat)) = aa(nat,A,F1,Nat) ).

% enat.simps(4)
tff(fact_7692_enat__ord__simps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(Mb)),extended_enat2(Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% enat_ord_simps(2)
tff(fact_7693_plus__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),extended_enat2(Mb)),extended_enat2(Nb)) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ).

% plus_enat_simps(1)
tff(fact_7694_enat__ord__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_enat2(Mb)),extended_enat2(Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% enat_ord_simps(1)
tff(fact_7695_idiff__enat__0__right,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Nb),extended_enat2(zero_zero(nat))) = Nb ).

% idiff_enat_0_right
tff(fact_7696_idiff__enat__0,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_enat2(zero_zero(nat))),Nb) = extended_enat2(zero_zero(nat)) ).

% idiff_enat_0
tff(fact_7697_idiff__enat__enat,axiom,
    ! [A3: nat,B2: nat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_enat2(A3)),extended_enat2(B2)) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) ).

% idiff_enat_enat
tff(fact_7698_times__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_enat2(Mb)),extended_enat2(Nb)) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) ).

% times_enat_simps(1)
tff(fact_7699_max__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),extended_enat2(Mb)),extended_enat2(Nb)) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),Nb)) ).

% max_enat_simps(1)
tff(fact_7700_min__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),extended_enat2(Mb)),extended_enat2(Nb)) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)) ).

% min_enat_simps(1)
tff(fact_7701_numeral__less__enat__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),Mb)),extended_enat2(Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),Mb)),Nb) ) ).

% numeral_less_enat_iff
tff(fact_7702_Suc__ile__eq,axiom,
    ! [Mb: nat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_enat2(aa(nat,nat,suc,Mb))),Nb)
    <=> aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(Mb)),Nb) ) ).

% Suc_ile_eq
tff(fact_7703_enat__ile,axiom,
    ! [Nb: extended_enat,Mb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Nb),extended_enat2(Mb))
     => ? [K3: nat] : Nb = extended_enat2(K3) ) ).

% enat_ile
tff(fact_7704_finite__enat__bounded,axiom,
    ! [A4: set(extended_enat),Nb: nat] :
      ( ! [Y4: extended_enat] :
          ( member(extended_enat,Y4,A4)
         => aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Y4),extended_enat2(Nb)) )
     => finite_finite(extended_enat,A4) ) ).

% finite_enat_bounded
tff(fact_7705_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,semiring_1_Nats(A))
         => ( member(A,B2,semiring_1_Nats(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
             => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2),semiring_1_Nats(A)) ) ) ) ) ).

% Nats_diff
tff(fact_7706_enat__1__iff_I2_J,axiom,
    ! [Xb: nat] :
      ( ( one_one(extended_enat) = extended_enat2(Xb) )
    <=> ( Xb = one_one(nat) ) ) ).

% enat_1_iff(2)
tff(fact_7707_enat__1__iff_I1_J,axiom,
    ! [Xb: nat] :
      ( ( extended_enat2(Xb) = one_one(extended_enat) )
    <=> ( Xb = one_one(nat) ) ) ).

% enat_1_iff(1)
tff(fact_7708_one__enat__def,axiom,
    one_one(extended_enat) = extended_enat2(one_one(nat)) ).

% one_enat_def
tff(fact_7709_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: A] :
          ( member(A,Xb,semiring_1_Nats(A))
         => ~ ! [N: nat] : Xb != aa(nat,A,semiring_1_of_nat(A),N) ) ) ).

% Nats_cases
tff(fact_7710_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: A,P: fun(A,$o)] :
          ( member(A,Xb,semiring_1_Nats(A))
         => ( ! [N: nat] : aa(A,$o,P,aa(nat,A,semiring_1_of_nat(A),N))
           => aa(A,$o,P,Xb) ) ) ) ).

% Nats_induct
tff(fact_7711_of__nat__in__Nats,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Nb),semiring_1_Nats(A)) ) ).

% of_nat_in_Nats
tff(fact_7712_of__nat__eq__enat,axiom,
    ! [Nb: nat] : aa(nat,extended_enat,semiring_1_of_nat(extended_enat),Nb) = extended_enat2(Nb) ).

% of_nat_eq_enat
tff(fact_7713_Nats__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [W: num] : member(A,aa(num,A,numeral_numeral(A),W),semiring_1_Nats(A)) ) ).

% Nats_numeral
tff(fact_7714_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,semiring_1_Nats(A))
         => ( member(A,B2,semiring_1_Nats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),semiring_1_Nats(A)) ) ) ) ).

% Nats_add
tff(fact_7715_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,one_one(A),semiring_1_Nats(A)) ) ).

% Nats_1
tff(fact_7716_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( member(A,A3,semiring_1_Nats(A))
         => ( member(A,B2,semiring_1_Nats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),semiring_1_Nats(A)) ) ) ) ).

% Nats_mult
tff(fact_7717_numeral__eq__enat,axiom,
    ! [K: num] : aa(num,extended_enat,numeral_numeral(extended_enat),K) = extended_enat2(aa(num,nat,numeral_numeral(nat),K)) ).

% numeral_eq_enat
tff(fact_7718_enat__iless,axiom,
    ! [Nb: extended_enat,Mb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Nb),extended_enat2(Mb))
     => ? [K3: nat] : Nb = extended_enat2(K3) ) ).

% enat_iless
tff(fact_7719_chain__incr,axiom,
    ! [A: $tType,Y6: fun(A,extended_enat),K: nat] :
      ( ! [I2: A] :
        ? [J4: A] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(A,extended_enat,Y6,I2)),aa(A,extended_enat,Y6,J4))
     => ? [J2: A] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(K)),aa(A,extended_enat,Y6,J2)) ) ).

% chain_incr
tff(fact_7720_less__enatE,axiom,
    ! [Nb: extended_enat,Mb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Nb),extended_enat2(Mb))
     => ~ ! [K3: nat] :
            ( ( Nb = extended_enat2(K3) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Mb) ) ) ).

% less_enatE
tff(fact_7721_zero__enat__def,axiom,
    zero_zero(extended_enat) = extended_enat2(zero_zero(nat)) ).

% zero_enat_def
tff(fact_7722_enat__0__iff_I1_J,axiom,
    ! [Xb: nat] :
      ( ( extended_enat2(Xb) = zero_zero(extended_enat) )
    <=> ( Xb = zero_zero(nat) ) ) ).

% enat_0_iff(1)
tff(fact_7723_enat__0__iff_I2_J,axiom,
    ! [Xb: nat] :
      ( ( zero_zero(extended_enat) = extended_enat2(Xb) )
    <=> ( Xb = zero_zero(nat) ) ) ).

% enat_0_iff(2)
tff(fact_7724_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,zero_zero(A),semiring_1_Nats(A)) ) ).

% Nats_0
tff(fact_7725_iadd__le__enat__iff,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Nb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xb),Y)),extended_enat2(Nb))
    <=> ? [Y8: nat,X10: nat] :
          ( ( Xb = extended_enat2(X10) )
          & ( Y = extended_enat2(Y8) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X10),Y8)),Nb) ) ) ).

% iadd_le_enat_iff
tff(fact_7726_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = image(nat,A,semiring_1_of_nat(A),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_7727_elimnum,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)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))) = vEBT_Node(Info,Dega,TreeLista,Summarya) ) ) ).

% elimnum
tff(fact_7728_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extended_enat2(L)) = vEBT_Node(Info,Dega,take(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_agi(nat,fun(vEBT_VEBT,vEBT_VEBT),Dega),TreeLista)),vEBT_VEBT_elim_dead(Summarya,extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_7729_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xb,Xaa) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A5),(B3)) )
           => ( Y != vEBT_Leaf((A5),(B3)) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
             => ( ( Xaa = extend4730790105801354508finity(extended_enat) )
               => ( Y != vEBT_Node(Info2,Deg,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_agi(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg),TreeList),vEBT_VEBT_elim_dead(Summary,extend4730790105801354508finity(extended_enat))) ) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
               => ! [L3: nat] :
                    ( ( Xaa = extended_enat2(L3) )
                   => ( Y != vEBT_Node(Info2,Deg,take(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_agi(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg),TreeList)),vEBT_VEBT_elim_dead(Summary,extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_7730_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extend4730790105801354508finity(extended_enat)) = vEBT_Node(Info,Dega,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_agi(nat,fun(vEBT_VEBT,vEBT_VEBT),Dega),TreeLista),vEBT_VEBT_elim_dead(Summarya,extend4730790105801354508finity(extended_enat))) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_7731_elimcomplete,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)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extend4730790105801354508finity(extended_enat)) = vEBT_Node(Info,Dega,TreeLista,Summarya) ) ) ).

% elimcomplete
tff(fact_7732_not__infinity__eq,axiom,
    ! [Xb: extended_enat] :
      ( ( Xb != extend4730790105801354508finity(extended_enat) )
    <=> ? [I4: nat] : Xb = extended_enat2(I4) ) ).

% not_infinity_eq
tff(fact_7733_not__enat__eq,axiom,
    ! [Xb: extended_enat] :
      ( ! [Y3: nat] : Xb != extended_enat2(Y3)
    <=> ( Xb = extend4730790105801354508finity(extended_enat) ) ) ).

% not_enat_eq
tff(fact_7734_enat__ord__simps_I4_J,axiom,
    ! [Q2: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Q2),extend4730790105801354508finity(extended_enat))
    <=> ( Q2 != extend4730790105801354508finity(extended_enat) ) ) ).

% enat_ord_simps(4)
tff(fact_7735_enat__ord__simps_I6_J,axiom,
    ! [Q2: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extend4730790105801354508finity(extended_enat)),Q2) ).

% enat_ord_simps(6)
tff(fact_7736_plus__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),extend4730790105801354508finity(extended_enat)),Q2) = extend4730790105801354508finity(extended_enat) ).

% plus_enat_simps(2)
tff(fact_7737_plus__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Q2),extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ).

% plus_enat_simps(3)
tff(fact_7738_enat__ord__code_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Q2),extend4730790105801354508finity(extended_enat)) ).

% enat_ord_code(3)
tff(fact_7739_enat__ord__simps_I5_J,axiom,
    ! [Q2: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extend4730790105801354508finity(extended_enat)),Q2)
    <=> ( Q2 = extend4730790105801354508finity(extended_enat) ) ) ).

% enat_ord_simps(5)
tff(fact_7740_idiff__infinity,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extend4730790105801354508finity(extended_enat)),Nb) = extend4730790105801354508finity(extended_enat) ).

% idiff_infinity
tff(fact_7741_enat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(nat,A),F22: A] : aa(extended_enat,A,aa(A,fun(extended_enat,A),aa(fun(nat,A),fun(A,fun(extended_enat,A)),extended_case_enat(A),F1),F22),extend4730790105801354508finity(extended_enat)) = F22 ).

% enat.simps(5)
tff(fact_7742_times__enat__simps_I2_J,axiom,
    aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ).

% times_enat_simps(2)
tff(fact_7743_max__enat__simps_I4_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Q2),extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ).

% max_enat_simps(4)
tff(fact_7744_max__enat__simps_I5_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),extend4730790105801354508finity(extended_enat)),Q2) = extend4730790105801354508finity(extended_enat) ).

% max_enat_simps(5)
tff(fact_7745_min__enat__simps_I5_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),extend4730790105801354508finity(extended_enat)),Q2) = Q2 ).

% min_enat_simps(5)
tff(fact_7746_min__enat__simps_I4_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),Q2),extend4730790105801354508finity(extended_enat)) = Q2 ).

% min_enat_simps(4)
tff(fact_7747_idiff__self,axiom,
    ! [Nb: extended_enat] :
      ( ( Nb != extend4730790105801354508finity(extended_enat) )
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Nb),Nb) = zero_zero(extended_enat) ) ) ).

% idiff_self
tff(fact_7748_add__diff__cancel__enat,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ( Xb != extend4730790105801354508finity(extended_enat) )
     => ( 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)),Xb) = Y ) ) ).

% add_diff_cancel_enat
tff(fact_7749_idiff__infinity__right,axiom,
    ! [A3: nat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_enat2(A3)),extend4730790105801354508finity(extended_enat)) = zero_zero(extended_enat) ).

% idiff_infinity_right
tff(fact_7750_times__enat__simps_I3_J,axiom,
    ! [Nb: nat] :
      aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(Nb)) = $ite(Nb = zero_zero(nat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)) ).

% times_enat_simps(3)
tff(fact_7751_times__enat__simps_I4_J,axiom,
    ! [Mb: nat] :
      aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_enat2(Mb)),extend4730790105801354508finity(extended_enat)) = $ite(Mb = zero_zero(nat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)) ).

% times_enat_simps(4)
tff(fact_7752_Sup__enat__def,axiom,
    ! [A4: set(extended_enat)] :
      aa(set(extended_enat),extended_enat,complete_Sup_Sup(extended_enat),A4) = $ite(
        A4 = bot_bot(set(extended_enat)),
        zero_zero(extended_enat),
        $ite(finite_finite(extended_enat,A4),aa(set(extended_enat),extended_enat,lattic643756798349783984er_Max(extended_enat),A4),extend4730790105801354508finity(extended_enat)) ) ).

% Sup_enat_def
tff(fact_7753_enat__add__left__cancel__le,axiom,
    ! [A3: extended_enat,B2: extended_enat,C2: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),B2)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),C2))
    <=> ( ( A3 = extend4730790105801354508finity(extended_enat) )
        | aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),B2),C2) ) ) ).

% enat_add_left_cancel_le
tff(fact_7754_enat__ord__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Q2),extend4730790105801354508finity(extended_enat)) ).

% enat_ord_simps(3)
tff(fact_7755_Inf__enat__def,axiom,
    ! [A4: set(extended_enat)] :
      aa(set(extended_enat),extended_enat,complete_Inf_Inf(extended_enat),A4) = $ite(A4 = bot_bot(set(extended_enat)),extend4730790105801354508finity(extended_enat),ord_Least(extended_enat,aTP_Lamp_agj(set(extended_enat),fun(extended_enat,$o),A4))) ).

% Inf_enat_def
tff(fact_7756_numeral__ne__infinity,axiom,
    ! [K: num] : aa(num,extended_enat,numeral_numeral(extended_enat),K) != extend4730790105801354508finity(extended_enat) ).

% numeral_ne_infinity
tff(fact_7757_infinity__ne__i1,axiom,
    extend4730790105801354508finity(extended_enat) != one_one(extended_enat) ).

% infinity_ne_i1
tff(fact_7758_enat__add__left__cancel,axiom,
    ! [A3: extended_enat,B2: extended_enat,C2: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),B2) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),C2) )
    <=> ( ( A3 = extend4730790105801354508finity(extended_enat) )
        | ( B2 = C2 ) ) ) ).

% enat_add_left_cancel
tff(fact_7759_plus__eq__infty__iff__enat,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) = extend4730790105801354508finity(extended_enat) )
    <=> ( ( Mb = extend4730790105801354508finity(extended_enat) )
        | ( Nb = extend4730790105801354508finity(extended_enat) ) ) ) ).

% plus_eq_infty_iff_enat
tff(fact_7760_enat__add__left__cancel__less,axiom,
    ! [A3: extended_enat,B2: extended_enat,C2: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),B2)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),C2))
    <=> ( ( A3 != extend4730790105801354508finity(extended_enat) )
        & aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),B2),C2) ) ) ).

% enat_add_left_cancel_less
tff(fact_7761_imult__is__infinity,axiom,
    ! [A3: extended_enat,B2: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),A3),B2) = extend4730790105801354508finity(extended_enat) )
    <=> ( ( ( A3 = extend4730790105801354508finity(extended_enat) )
          & ( B2 != zero_zero(extended_enat) ) )
        | ( ( B2 = extend4730790105801354508finity(extended_enat) )
          & ( A3 != zero_zero(extended_enat) ) ) ) ) ).

% imult_is_infinity
tff(fact_7762_infinity__ne__i0,axiom,
    extend4730790105801354508finity(extended_enat) != zero_zero(extended_enat) ).

% infinity_ne_i0
tff(fact_7763_top__enat__def,axiom,
    top_top(extended_enat) = extend4730790105801354508finity(extended_enat) ).

% top_enat_def
tff(fact_7764_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,extended_enat)] :
      ( ! [A5: $o,B3: $o,Uu2: extended_enat] : Xb != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Leaf((A5),(B3))),Uu2)
     => ( ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Node(Info2,Deg,TreeList,Summary)),extend4730790105801354508finity(extended_enat))
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,L3: nat] : Xb != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Node(Info2,Deg,TreeList,Summary)),extended_enat2(L3)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_7765_infinity__ilessE,axiom,
    ! [Mb: nat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(Mb)) ).

% infinity_ilessE
tff(fact_7766_less__infinityE,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Nb),extend4730790105801354508finity(extended_enat))
     => ~ ! [K3: nat] : Nb != extended_enat2(K3) ) ).

% less_infinityE
tff(fact_7767_enat__ord__code_I4_J,axiom,
    ! [Mb: nat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(Mb)),extend4730790105801354508finity(extended_enat)) ).

% enat_ord_code(4)
tff(fact_7768_enat__ex__split,axiom,
    ! [P: fun(extended_enat,$o)] :
      ( ? [X_1: extended_enat] : aa(extended_enat,$o,P,X_1)
    <=> ( aa(extended_enat,$o,P,extend4730790105801354508finity(extended_enat))
        | ? [X: nat] : aa(extended_enat,$o,P,extended_enat2(X)) ) ) ).

% enat_ex_split
tff(fact_7769_enat3__cases,axiom,
    ! [Y: extended_enat,Ya: extended_enat,Yb: extended_enat] :
      ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
       => ( ? [Nata: nat] : Ya = extended_enat2(Nata)
         => ! [Natb: nat] : Yb != extended_enat2(Natb) ) )
     => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
         => ( ? [Nata: nat] : Ya = extended_enat2(Nata)
           => ( Yb != extend4730790105801354508finity(extended_enat) ) ) )
       => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
           => ( ( Ya = extend4730790105801354508finity(extended_enat) )
             => ! [Nata: nat] : Yb != extended_enat2(Nata) ) )
         => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
             => ( ( Ya = extend4730790105801354508finity(extended_enat) )
               => ( Yb != extend4730790105801354508finity(extended_enat) ) ) )
           => ( ( ( Y = extend4730790105801354508finity(extended_enat) )
               => ( ? [Nat3: nat] : Ya = extended_enat2(Nat3)
                 => ! [Nata: nat] : Yb != extended_enat2(Nata) ) )
             => ( ( ( Y = extend4730790105801354508finity(extended_enat) )
                 => ( ? [Nat3: nat] : Ya = extended_enat2(Nat3)
                   => ( Yb != extend4730790105801354508finity(extended_enat) ) ) )
               => ( ( ( Y = extend4730790105801354508finity(extended_enat) )
                   => ( ( Ya = extend4730790105801354508finity(extended_enat) )
                     => ! [Nat3: nat] : Yb != extended_enat2(Nat3) ) )
                 => ~ ( ( Y = extend4730790105801354508finity(extended_enat) )
                     => ( ( Ya = extend4730790105801354508finity(extended_enat) )
                       => ( Yb != extend4730790105801354508finity(extended_enat) ) ) ) ) ) ) ) ) ) ) ).

% enat3_cases
tff(fact_7770_enat2__cases,axiom,
    ! [Y: extended_enat,Ya: extended_enat] :
      ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
       => ! [Nata: nat] : Ya != extended_enat2(Nata) )
     => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
         => ( Ya != extend4730790105801354508finity(extended_enat) ) )
       => ( ( ( Y = extend4730790105801354508finity(extended_enat) )
           => ! [Nat3: nat] : Ya != extended_enat2(Nat3) )
         => ~ ( ( Y = extend4730790105801354508finity(extended_enat) )
             => ( Ya != extend4730790105801354508finity(extended_enat) ) ) ) ) ) ).

% enat2_cases
tff(fact_7771_enat_Oexhaust,axiom,
    ! [Y: extended_enat] :
      ( ! [Nat3: nat] : Y != extended_enat2(Nat3)
     => ( Y = extend4730790105801354508finity(extended_enat) ) ) ).

% enat.exhaust
tff(fact_7772_enat_Odistinct_I1_J,axiom,
    ! [Nat: nat] : extended_enat2(Nat) != extend4730790105801354508finity(extended_enat) ).

% enat.distinct(1)
tff(fact_7773_enat_Odistinct_I2_J,axiom,
    ! [Nat: nat] : extend4730790105801354508finity(extended_enat) != extended_enat2(Nat) ).

% enat.distinct(2)
tff(fact_7774_infinity__ileE,axiom,
    ! [Mb: nat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(Mb)) ).

% infinity_ileE
tff(fact_7775_enat__ord__code_I5_J,axiom,
    ! [Nb: nat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(Nb)) ).

% enat_ord_code(5)
tff(fact_7776_imult__infinity,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Nb)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),Nb) = extend4730790105801354508finity(extended_enat) ) ) ).

% imult_infinity
tff(fact_7777_imult__infinity__right,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Nb)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),Nb),extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ) ) ).

% imult_infinity_right
tff(fact_7778_plus__enat__def,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) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),aa(fun(nat,extended_enat),fun(extended_enat,fun(extended_enat,extended_enat)),extended_case_enat(extended_enat),aTP_Lamp_agl(extended_enat,fun(nat,extended_enat),Nb)),extend4730790105801354508finity(extended_enat)),Mb) ).

% plus_enat_def
tff(fact_7779_diff__enat__def,axiom,
    ! [A3: extended_enat,B2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),A3),B2) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),aa(fun(nat,extended_enat),fun(extended_enat,fun(extended_enat,extended_enat)),extended_case_enat(extended_enat),aTP_Lamp_agn(extended_enat,fun(nat,extended_enat),B2)),extend4730790105801354508finity(extended_enat)),A3) ).

% diff_enat_def
tff(fact_7780_times__enat__def,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) = aa(extended_enat,extended_enat,
        aa(extended_enat,fun(extended_enat,extended_enat),aa(fun(nat,extended_enat),fun(extended_enat,fun(extended_enat,extended_enat)),extended_case_enat(extended_enat),aTP_Lamp_agp(extended_enat,fun(nat,extended_enat),Nb)),
          $ite(Nb = zero_zero(extended_enat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat))),
        Mb) ).

% times_enat_def
tff(fact_7781_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),Xb),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = vEBT_Leaf((A5),(B3)) )
               => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
               => ( ( Xaa = extend4730790105801354508finity(extended_enat) )
                 => ( ( Y = vEBT_Node(Info2,Deg,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_agi(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg),TreeList),vEBT_VEBT_elim_dead(Summary,extend4730790105801354508finity(extended_enat))) )
                   => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Node(Info2,Deg,TreeList,Summary)),extend4730790105801354508finity(extended_enat))) ) ) )
           => ~ ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
                 => ! [L3: nat] :
                      ( ( Xaa = extended_enat2(L3) )
                     => ( ( Y = vEBT_Node(Info2,Deg,take(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_agi(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg),TreeList)),vEBT_VEBT_elim_dead(Summary,extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) )
                       => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Node(Info2,Deg,TreeList,Summary)),extended_enat2(L3))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_7782_the__enat_Osimps,axiom,
    ! [Nb: nat] : extended_the_enat(extended_enat2(Nb)) = Nb ).

% the_enat.simps
tff(fact_7783_eSuc__Max,axiom,
    ! [A4: set(extended_enat)] :
      ( finite_finite(extended_enat,A4)
     => ( ( A4 != bot_bot(set(extended_enat)) )
       => ( aa(extended_enat,extended_enat,extended_eSuc,aa(set(extended_enat),extended_enat,lattic643756798349783984er_Max(extended_enat),A4)) = aa(set(extended_enat),extended_enat,lattic643756798349783984er_Max(extended_enat),image(extended_enat,extended_enat,extended_eSuc,A4)) ) ) ) ).

% eSuc_Max
tff(fact_7784_enat_Osimps_I7_J,axiom,
    ! [A: $tType,F1: fun(nat,A),F22: A] : aa(extended_enat,A,aa(A,fun(extended_enat,A),aa(fun(nat,A),fun(A,fun(extended_enat,A)),extended_rec_enat(A),F1),F22),extend4730790105801354508finity(extended_enat)) = F22 ).

% enat.simps(7)
tff(fact_7785_eSuc__inject,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( aa(extended_enat,extended_enat,extended_eSuc,Mb) = aa(extended_enat,extended_enat,extended_eSuc,Nb) )
    <=> ( Mb = Nb ) ) ).

% eSuc_inject
tff(fact_7786_eSuc__infinity,axiom,
    aa(extended_enat,extended_enat,extended_eSuc,extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ).

% eSuc_infinity
tff(fact_7787_eSuc__mono,axiom,
    ! [Nb: extended_enat,Mb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(extended_enat,extended_enat,extended_eSuc,Nb)),aa(extended_enat,extended_enat,extended_eSuc,Mb))
    <=> aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Nb),Mb) ) ).

% eSuc_mono
tff(fact_7788_eSuc__ile__mono,axiom,
    ! [Nb: extended_enat,Mb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(extended_enat,extended_enat,extended_eSuc,Nb)),aa(extended_enat,extended_enat,extended_eSuc,Mb))
    <=> aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Nb),Mb) ) ).

% eSuc_ile_mono
tff(fact_7789_eSuc__minus__eSuc,axiom,
    ! [Nb: extended_enat,Mb: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),aa(extended_enat,extended_enat,extended_eSuc,Nb)),aa(extended_enat,extended_enat,extended_eSuc,Mb)) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Nb),Mb) ).

% eSuc_minus_eSuc
tff(fact_7790_enat_Osimps_I6_J,axiom,
    ! [A: $tType,F1: fun(nat,A),F22: A,Nat: nat] : aa(extended_enat,A,aa(A,fun(extended_enat,A),aa(fun(nat,A),fun(A,fun(extended_enat,A)),extended_rec_enat(A),F1),F22),extended_enat2(Nat)) = aa(nat,A,F1,Nat) ).

% enat.simps(6)
tff(fact_7791_iless__eSuc0,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Nb),aa(extended_enat,extended_enat,extended_eSuc,zero_zero(extended_enat)))
    <=> ( Nb = zero_zero(extended_enat) ) ) ).

% iless_eSuc0
tff(fact_7792_eSuc__minus__1,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),aa(extended_enat,extended_enat,extended_eSuc,Nb)),one_one(extended_enat)) = Nb ).

% eSuc_minus_1
tff(fact_7793_eSuc__numeral,axiom,
    ! [K: num] : aa(extended_enat,extended_enat,extended_eSuc,aa(num,extended_enat,numeral_numeral(extended_enat),K)) = aa(num,extended_enat,numeral_numeral(extended_enat),aa(num,num,aa(num,fun(num,num),plus_plus(num),K),one2)) ).

% eSuc_numeral
tff(fact_7794_iless__Suc__eq,axiom,
    ! [Mb: nat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(Mb)),aa(extended_enat,extended_enat,extended_eSuc,Nb))
    <=> aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_enat2(Mb)),Nb) ) ).

% iless_Suc_eq
tff(fact_7795_ileI1,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Mb),Nb)
     => aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(extended_enat,extended_enat,extended_eSuc,Mb)),Nb) ) ).

% ileI1
tff(fact_7796_mono__eSuc,axiom,
    aa(fun(extended_enat,extended_enat),$o,order_mono(extended_enat,extended_enat),extended_eSuc) ).

% mono_eSuc
tff(fact_7797_sup__continuous__eSuc,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(A,extended_enat)] :
          ( order_sup_continuous(A,extended_enat,F2)
         => order_sup_continuous(A,extended_enat,aTP_Lamp_agq(fun(A,extended_enat),fun(A,extended_enat),F2)) ) ) ).

% sup_continuous_eSuc
tff(fact_7798_ile__eSuc,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Nb),aa(extended_enat,extended_enat,extended_eSuc,Nb)) ).

% ile_eSuc
tff(fact_7799_mult__eSuc,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),aa(extended_enat,extended_enat,extended_eSuc,Mb)),Nb) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Nb),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),Mb),Nb)) ).

% mult_eSuc
tff(fact_7800_mult__eSuc__right,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),aa(extended_enat,extended_enat,extended_eSuc,Nb)) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Mb),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),Mb),Nb)) ).

% mult_eSuc_right
tff(fact_7801_iadd__Suc__right,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),aa(extended_enat,extended_enat,extended_eSuc,Nb)) = aa(extended_enat,extended_enat,extended_eSuc,aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Mb),Nb)) ).

% iadd_Suc_right
tff(fact_7802_iadd__Suc,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),aa(extended_enat,extended_enat,extended_eSuc,Mb)),Nb) = aa(extended_enat,extended_enat,extended_eSuc,aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Mb),Nb)) ).

% iadd_Suc
tff(fact_7803_eSuc__max,axiom,
    ! [Xb: extended_enat,Y: extended_enat] : aa(extended_enat,extended_enat,extended_eSuc,aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Xb),Y)) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),aa(extended_enat,extended_enat,extended_eSuc,Xb)),aa(extended_enat,extended_enat,extended_eSuc,Y)) ).

% eSuc_max
tff(fact_7804_eSuc__plus__1,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,extended_enat,extended_eSuc,Nb) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Nb),one_one(extended_enat)) ).

% eSuc_plus_1
tff(fact_7805_plus__1__eSuc_I1_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),one_one(extended_enat)),Q2) = aa(extended_enat,extended_enat,extended_eSuc,Q2) ).

% plus_1_eSuc(1)
tff(fact_7806_plus__1__eSuc_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Q2),one_one(extended_enat)) = aa(extended_enat,extended_enat,extended_eSuc,Q2) ).

% plus_1_eSuc(2)
tff(fact_7807_i0__iless__eSuc,axiom,
    ! [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,extended_eSuc,Nb)) ).

% i0_iless_eSuc
tff(fact_7808_one__eSuc,axiom,
    one_one(extended_enat) = aa(extended_enat,extended_enat,extended_eSuc,zero_zero(extended_enat)) ).

% one_eSuc
tff(fact_7809_zero__ne__eSuc,axiom,
    ! [Nb: extended_enat] : zero_zero(extended_enat) != aa(extended_enat,extended_enat,extended_eSuc,Nb) ).

% zero_ne_eSuc
tff(fact_7810_not__eSuc__ilei0,axiom,
    ! [Nb: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(extended_enat,extended_enat,extended_eSuc,Nb)),zero_zero(extended_enat)) ).

% not_eSuc_ilei0
tff(fact_7811_eSuc__enat,axiom,
    ! [Nb: nat] : aa(extended_enat,extended_enat,extended_eSuc,extended_enat2(Nb)) = extended_enat2(aa(nat,nat,suc,Nb)) ).

% eSuc_enat
tff(fact_7812_eSuc__enat__iff,axiom,
    ! [Xb: extended_enat,Y: nat] :
      ( ( aa(extended_enat,extended_enat,extended_eSuc,Xb) = extended_enat2(Y) )
    <=> ? [N4: nat] :
          ( ( Y = aa(nat,nat,suc,N4) )
          & ( Xb = extended_enat2(N4) ) ) ) ).

% eSuc_enat_iff
tff(fact_7813_enat__eSuc__iff,axiom,
    ! [Y: nat,Xb: extended_enat] :
      ( ( extended_enat2(Y) = aa(extended_enat,extended_enat,extended_eSuc,Xb) )
    <=> ? [N4: nat] :
          ( ( Y = aa(nat,nat,suc,N4) )
          & ( extended_enat2(N4) = Xb ) ) ) ).

% enat_eSuc_iff
tff(fact_7814_eSuc__Sup,axiom,
    ! [A4: set(extended_enat)] :
      ( ( A4 != bot_bot(set(extended_enat)) )
     => ( aa(extended_enat,extended_enat,extended_eSuc,aa(set(extended_enat),extended_enat,complete_Sup_Sup(extended_enat),A4)) = aa(set(extended_enat),extended_enat,complete_Sup_Sup(extended_enat),image(extended_enat,extended_enat,extended_eSuc,A4)) ) ) ).

% eSuc_Sup
tff(fact_7815_case__enat__def,axiom,
    ! [A: $tType] : extended_case_enat(A) = extended_rec_enat(A) ).

% case_enat_def
tff(fact_7816_eSuc__def,axiom,
    ! [I: extended_enat] : aa(extended_enat,extended_enat,extended_eSuc,I) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),aa(fun(nat,extended_enat),fun(extended_enat,fun(extended_enat,extended_enat)),extended_case_enat(extended_enat),aTP_Lamp_agr(nat,extended_enat)),extend4730790105801354508finity(extended_enat)),I) ).

% eSuc_def
tff(fact_7817_rec__enat__def,axiom,
    ! [A: $tType,X5: fun(nat,A),Xa: A,Xb2: extended_enat] : aa(extended_enat,A,aa(A,fun(extended_enat,A),aa(fun(nat,A),fun(A,fun(extended_enat,A)),extended_rec_enat(A),X5),Xa),Xb2) = the(A,extend4933016492236175606t_enat(A,X5,Xa,Xb2)) ).

% rec_enat_def
tff(fact_7818_less__than__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Y),less_than)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y) ) ).

% less_than_iff
tff(fact_7819_lenlex__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R)),aTP_Lamp_ags(list(A),product_prod(nat,list(A)))) ).

% lenlex_def
tff(fact_7820_mlex__prod__def,axiom,
    ! [A: $tType,F2: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F2,R2) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R2),aTP_Lamp_agt(fun(A,nat),fun(A,product_prod(nat,A)),F2)) ).

% mlex_prod_def
tff(fact_7821_in__inv__image,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: A,R: set(product_prod(B,B)),F2: fun(A,B)] :
      ( 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),inv_image(B,A,R,F2))
    <=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)),R) ) ).

% in_inv_image
tff(fact_7822_inv__image__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(B,B)),F2: fun(A,B)] : inv_image(B,A,R,F2) = aa(fun(product_prod(A,A),$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,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_agu(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),R),F2))) ).

% inv_image_def
tff(fact_7823_has__vector__derivative__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F6: real,Xb: real,S: set(real),G: fun(real,A),G5: A] :
          ( has_field_derivative(real,F2,F6,topolo174197925503356063within(real,Xb,S))
         => ( has_ve8173657378732805170vative(A,G,G5,topolo174197925503356063within(real,Xb,S))
           => has_ve8173657378732805170vative(A,aa(fun(real,A),fun(real,A),aTP_Lamp_agv(fun(real,real),fun(fun(real,A),fun(real,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,F2,Xb)),G5)),aa(A,A,real_V8093663219630862766scaleR(A,F6),aa(real,A,G,Xb))),topolo174197925503356063within(real,Xb,S)) ) ) ) ).

% has_vector_derivative_scaleR
tff(fact_7824_Quotient__real,axiom,
    quotient(fun(nat,rat),real,realrel,real2,rep_real2,cr_real) ).

% Quotient_real
tff(fact_7825_has__vector__derivative__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(real,A),Xb: A,F4: filter(real),A3: A] :
          ( has_ve8173657378732805170vative(A,F2,Xb,F4)
         => has_ve8173657378732805170vative(A,aa(A,fun(real,A),aTP_Lamp_agw(fun(real,A),fun(A,fun(real,A)),F2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),A3),F4) ) ) ).

% has_vector_derivative_mult_left
tff(fact_7826_has__vector__derivative__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(real,A),Xb: A,F4: filter(real),A3: A] :
          ( has_ve8173657378732805170vative(A,F2,Xb,F4)
         => has_ve8173657378732805170vative(A,aa(A,fun(real,A),aTP_Lamp_agx(fun(real,A),fun(A,fun(real,A)),F2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Xb),F4) ) ) ).

% has_vector_derivative_mult_right
tff(fact_7827_has__vector__derivative__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(real,A),Xb: A,F4: filter(real),A3: A] :
          ( has_ve8173657378732805170vative(A,F2,Xb,F4)
         => has_ve8173657378732805170vative(A,aa(A,fun(real,A),aTP_Lamp_agy(fun(real,A),fun(A,fun(real,A)),F2),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),A3),F4) ) ) ).

% has_vector_derivative_divide
tff(fact_7828_has__vector__derivative__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,A),F6: A,Net: filter(real),G: fun(real,A),G5: A] :
          ( has_ve8173657378732805170vative(A,F2,F6,Net)
         => ( has_ve8173657378732805170vative(A,G,G5,Net)
           => has_ve8173657378732805170vative(A,aa(fun(real,A),fun(real,A),aTP_Lamp_agz(fun(real,A),fun(fun(real,A),fun(real,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F6),G5),Net) ) ) ) ).

% has_vector_derivative_add
tff(fact_7829_has__vector__derivative__add__const,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(real,A),Z: A,F6: A,Net: filter(real)] :
          ( has_ve8173657378732805170vative(A,aa(A,fun(real,A),aTP_Lamp_aha(fun(real,A),fun(A,fun(real,A)),G),Z),F6,Net)
        <=> has_ve8173657378732805170vative(A,G,F6,Net) ) ) ).

% has_vector_derivative_add_const
tff(fact_7830_has__vector__derivative__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(real,A),F6: A,Xb: real,S: set(real),G: fun(real,A),G5: A] :
          ( has_ve8173657378732805170vative(A,F2,F6,topolo174197925503356063within(real,Xb,S))
         => ( has_ve8173657378732805170vative(A,G,G5,topolo174197925503356063within(real,Xb,S))
           => has_ve8173657378732805170vative(A,aa(fun(real,A),fun(real,A),aTP_Lamp_ahb(fun(real,A),fun(fun(real,A),fun(real,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,F2,Xb)),G5)),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(real,A,G,Xb))),topolo174197925503356063within(real,Xb,S)) ) ) ) ).

% has_vector_derivative_mult
tff(fact_7831_bounded__bilinear_Ohas__vector__derivative,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(real,A),F6: A,Xb: real,S: set(real),G: fun(real,B),G5: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( has_ve8173657378732805170vative(A,F2,F6,topolo174197925503356063within(real,Xb,S))
           => ( has_ve8173657378732805170vative(B,G,G5,topolo174197925503356063within(real,Xb,S))
             => has_ve8173657378732805170vative(C,aa(fun(real,B),fun(real,C),aa(fun(real,A),fun(fun(real,B),fun(real,C)),aTP_Lamp_ahc(fun(A,fun(B,C)),fun(fun(real,A),fun(fun(real,B),fun(real,C))),Prod),F2),G),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,aa(real,A,F2,Xb)),G5)),aa(B,C,aa(A,fun(B,C),Prod,F6),aa(real,B,G,Xb))),topolo174197925503356063within(real,Xb,S)) ) ) ) ) ).

% bounded_bilinear.has_vector_derivative
tff(fact_7832_aboveS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_aboveS(A,R,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ahd(set(product_prod(A,A)),fun(A,fun(A,$o)),R),A3)) ).

% aboveS_def
tff(fact_7833_is__singleton__altdef,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_singleton(A,A4)
    <=> ( aa(set(A),nat,finite_card(A),A4) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_7834_relInvImage__def,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(B,B)),F2: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A4,R2,F2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),$o),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o)),aTP_Lamp_ahe(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),A4),R2),F2)) ).

% relInvImage_def
tff(fact_7835_prod__list__def,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( groups5270119922927024881d_list(A) = groups_monoid_F(A,times_times(A),one_one(A)) ) ) ).

% prod_list_def
tff(fact_7836_sum__list__def,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A) = groups_monoid_F(A,plus_plus(A),zero_zero(A)) ) ) ).

% sum_list_def
tff(fact_7837_ZfunD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),R: real] :
          ( zfun(A,B,F2,F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => eventually(A,aa(real,fun(A,$o),aTP_Lamp_ahf(fun(A,B),fun(real,fun(A,$o)),F2),R),F4) ) ) ) ).

% ZfunD
tff(fact_7838_ZfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => eventually(A,aa(real,fun(A,$o),aTP_Lamp_ahf(fun(A,B),fun(real,fun(A,$o)),F2),R3),F4) )
         => zfun(A,B,F2,F4) ) ) ).

% ZfunI
tff(fact_7839_Zfun__add,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( zfun(A,B,F2,F4)
         => ( zfun(A,B,G,F4)
           => zfun(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ww(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% Zfun_add
tff(fact_7840_Zfun__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),A3: B] :
          ( zfun(A,B,F2,F4)
         => zfun(A,B,aa(B,fun(A,B),aTP_Lamp_tm(fun(A,B),fun(B,fun(A,B)),F2),A3),F4) ) ) ).

% Zfun_mult_right
tff(fact_7841_Zfun__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),A3: B] :
          ( zfun(A,B,F2,F4)
         => zfun(A,B,aa(B,fun(A,B),aTP_Lamp_tl(fun(A,B),fun(B,fun(A,B)),F2),A3),F4) ) ) ).

% Zfun_mult_left
tff(fact_7842_Zfun__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( zfun(A,B,F2,F4)
         => ( zfun(A,B,G,F4)
           => zfun(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% Zfun_mult
tff(fact_7843_Zfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( zfun(A,B,F2,F4)
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => eventually(A,aa(real,fun(A,$o),aTP_Lamp_ahf(fun(A,B),fun(real,fun(A,$o)),F2),R5),F4) ) ) ) ).

% Zfun_def
tff(fact_7844_not__in__connected__cases,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),Xb: A] :
          ( topolo1966860045006549960nected(A,S3)
         => ( ~ member(A,Xb,S3)
           => ( ( S3 != bot_bot(set(A)) )
             => ( ( condit941137186595557371_above(A,S3)
                 => ~ ! [Y5: A] :
                        ( member(A,Y5,S3)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),Xb) ) )
               => ~ ( condit1013018076250108175_below(A,S3)
                   => ~ ! [Y5: A] :
                          ( member(A,Y5,S3)
                         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y5) ) ) ) ) ) ) ) ).

% not_in_connected_cases
tff(fact_7845_pair__in__swap__image,axiom,
    ! [A: $tType,B: $tType,Y: A,Xb: B,A4: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Xb),image(product_prod(B,A),product_prod(A,B),product_swap(B,A),A4))
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y),A4) ) ).

% pair_in_swap_image
tff(fact_7846_swap__simp,axiom,
    ! [A: $tType,B: $tType,Xb: B,Y: A] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Xb) ).

% swap_simp
tff(fact_7847_connectedD__interval,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [U2: set(A),Xb: A,Y: A,Z: A] :
          ( topolo1966860045006549960nected(A,U2)
         => ( member(A,Xb,U2)
           => ( member(A,Y,U2)
             => ( aa(A,$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)
                 => member(A,Z,U2) ) ) ) ) ) ) ).

% connectedD_interval
tff(fact_7848_connectedI__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U2: set(A)] :
          ( ! [X4: A,Y4: A,Z3: A] :
              ( member(A,X4,U2)
             => ( member(A,Y4,U2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),Y4)
                   => member(A,Z3,U2) ) ) ) )
         => topolo1966860045006549960nected(A,U2) ) ) ).

% connectedI_interval
tff(fact_7849_connected__iff__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U2: set(A)] :
          ( topolo1966860045006549960nected(A,U2)
        <=> ! [X: A] :
              ( member(A,X,U2)
             => ! [Xa2: A] :
                  ( member(A,Xa2,U2)
                 => ! [Z2: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
                     => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Xa2)
                       => member(A,Z2,U2) ) ) ) ) ) ) ).

% connected_iff_interval
tff(fact_7850_prod_Oswap__def,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(B,A)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),P2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(B,A),A,product_snd(B,A),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2)) ).

% prod.swap_def
tff(fact_7851_gfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F2),top_top(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A)) )
           => ( complete_lattice_gfp(A,F2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A)) ) ) ) ) ).

% gfp_Kleene_iter
tff(fact_7852_strict__sorted__equal__Uniq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : uniq(list(A),aTP_Lamp_ahg(set(A),fun(list(A),$o),A4)) ) ).

% strict_sorted_equal_Uniq
tff(fact_7853_gfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),Xb: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F4)
         => ( ( aa(A,A,F4,Xb) = Xb )
           => ( ! [Z3: A] :
                  ( ( aa(A,A,F4,Z3) = Z3 )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),Xb) )
             => ( complete_lattice_gfp(A,F4) = Xb ) ) ) ) ) ).

% gfp_eqI
tff(fact_7854_gfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A)] : complete_lattice_gfp(A,F2) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ahh(fun(A,A),fun(A,$o),F2))) ) ).

% gfp_def
tff(fact_7855_gfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),G: fun(A,A)] :
          ( ! [Z7: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,Z7)),aa(A,A,G,Z7))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_gfp(A,F2)),complete_lattice_gfp(A,G)) ) ) ).

% gfp_mono
tff(fact_7856_gfp__upperbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X6: A,F2: fun(A,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),aa(A,A,F2,X6))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),complete_lattice_gfp(A,F2)) ) ) ).

% gfp_upperbound
tff(fact_7857_gfp__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),X6: A] :
          ( ! [U3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U3),aa(A,A,F2,U3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U3),X6) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_gfp(A,F2)),X6) ) ) ).

% gfp_least
tff(fact_7858_gfp__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,fun(A,A))] :
          ( ! [X4: A,Y4: A,W2: A,Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X4),W2)),aa(A,A,aa(A,fun(A,A),F2,Y4),Z3)) ) )
         => ( complete_lattice_gfp(A,aTP_Lamp_ahi(fun(A,fun(A,A)),fun(A,A),F2)) = complete_lattice_gfp(A,aTP_Lamp_afq(fun(A,fun(A,A)),fun(A,A),F2)) ) ) ) ).

% gfp_gfp
tff(fact_7859_coinduct__lemma,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X6: A,F2: fun(A,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F2))))
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F2))),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F2)))) ) ) ) ).

% coinduct_lemma
tff(fact_7860_def__coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: A,F2: fun(A,A),X6: A] :
          ( ( A4 = complete_lattice_gfp(A,F2) )
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),A4)))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),A4) ) ) ) ) ).

% def_coinduct
tff(fact_7861_coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),X6: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F2))))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),complete_lattice_gfp(A,F2)) ) ) ) ).

% coinduct
tff(fact_7862_gfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: fun(A,$o)] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( ! [S4: A] :
                ( aa(A,$o,P,S4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_gfp(A,F2)),S4)
                 => aa(A,$o,P,aa(A,A,F2,S4)) ) )
           => ( ! [M7: set(A)] :
                  ( ! [X5: A] :
                      ( member(A,X5,M7)
                     => aa(A,$o,P,X5) )
                 => aa(A,$o,P,aa(set(A),A,complete_Inf_Inf(A),M7)) )
             => aa(A,$o,P,complete_lattice_gfp(A,F2)) ) ) ) ) ).

% gfp_ordinal_induct
tff(fact_7863_gfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Nb: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( complete_lattice_gfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2)) = complete_lattice_gfp(A,F2) ) ) ) ).

% gfp_funpow
tff(fact_7864_lfp__le__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A)] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),complete_lattice_gfp(A,F2)) ) ) ).

% lfp_le_gfp
tff(fact_7865_gfp__transfer__bounded,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [P: fun(A,$o),F2: fun(A,A),Alpha: fun(A,B),G: fun(B,B)] :
          ( aa(A,$o,P,aa(A,A,F2,top_top(A)))
         => ( ! [X4: A] :
                ( aa(A,$o,P,X4)
               => aa(A,$o,P,aa(A,A,F2,X4)) )
           => ( ! [M7: fun(nat,A)] :
                  ( order_antimono(nat,A,M7)
                 => ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M7,I3))
                   => aa(A,$o,P,aa(set(A),A,complete_Inf_Inf(A),image(nat,A,M7,top_top(set(nat))))) ) )
             => ( ! [M7: fun(nat,A)] :
                    ( order_antimono(nat,A,M7)
                   => ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M7,I3))
                     => ( aa(A,B,Alpha,aa(set(A),A,complete_Inf_Inf(A),image(nat,A,M7,top_top(set(nat))))) = aa(set(B),B,complete_Inf_Inf(B),image(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_afs(fun(A,B),fun(fun(nat,A),fun(nat,B)),Alpha),M7),top_top(set(nat)))) ) ) )
               => ( order_inf_continuous(A,A,F2)
                 => ( order_inf_continuous(B,B,G)
                   => ( ! [X4: A] :
                          ( aa(A,$o,P,X4)
                         => ( aa(A,B,Alpha,aa(A,A,F2,X4)) = aa(B,B,G,aa(A,B,Alpha,X4)) ) )
                     => ( ! [X4: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,G,X4)),aa(A,B,Alpha,aa(A,A,F2,top_top(A))))
                       => ( aa(A,B,Alpha,complete_lattice_gfp(A,F2)) = complete_lattice_gfp(B,G) ) ) ) ) ) ) ) ) ) ) ).

% gfp_transfer_bounded
tff(fact_7866_Zorns__po__lemma,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,field2(A,R),R)
     => ( ! [C7: set(A)] :
            ( member(set(A),C7,chains(A,R))
           => ? [X5: A] :
                ( member(A,X5,field2(A,R))
                & ! [Xa3: A] :
                    ( member(A,Xa3,C7)
                   => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5),R) ) ) )
       => ? [X4: A] :
            ( member(A,X4,field2(A,R))
            & ! [Xa: A] :
                ( member(A,Xa,field2(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),X4),Xa),R)
                 => ( Xa = X4 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_7867_list__ex__length,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( list_ex(A,P,Xs)
    <=> ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),N4)) ) ) ).

% list_ex_length
tff(fact_7868_and_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => semilattice_neutr(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.semilattice_neutr_axioms
tff(fact_7869_max__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.semilattice_neutr_axioms
tff(fact_7870_or_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => semilattice_neutr(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.semilattice_neutr_axioms
tff(fact_7871_cclfp__lowerbound,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(A,A),A4: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,A4)),A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),order_532582986084564980_cclfp(A,F2)),A4) ) ) ) ).

% cclfp_lowerbound
tff(fact_7872_internal__case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(B,fun(C,A)),A3: B,B2: C] : produc5280177257484947105e_prod(B,C,A,C2,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),C2,A3),B2) ).

% internal_case_prod_conv
tff(fact_7873_card__Un__disjnt,axiom,
    ! [A: $tType,A4: set(A),B4: set(A)] :
      ( finite_finite(A,A4)
     => ( finite_finite(A,B4)
       => ( disjnt(A,A4,B4)
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ) ).

% card_Un_disjnt
tff(fact_7874_Range__insert,axiom,
    ! [A: $tType,B: $tType,A3: B,B2: A,R: set(product_prod(B,A))] : range(B,A,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),A3),B2)),R)) = aa(set(A),set(A),insert(A,B2),range(B,A,R)) ).

% Range_insert
tff(fact_7875_Range_Ocases,axiom,
    ! [A: $tType,B: $tType,A3: A,R: set(product_prod(B,A))] :
      ( member(A,A3,range(B,A,R))
     => ~ ! [A5: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),A3),R) ) ).

% Range.cases
tff(fact_7876_Range_Osimps,axiom,
    ! [A: $tType,B: $tType,A3: A,R: set(product_prod(B,A))] :
      ( member(A,A3,range(B,A,R))
    <=> ? [A6: B,B5: A] :
          ( ( A3 = B5 )
          & member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A6),B5),R) ) ) ).

% Range.simps
tff(fact_7877_Range_Ointros,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(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),A3),B2),R)
     => member(B,B2,range(A,B,R)) ) ).

% Range.intros
tff(fact_7878_RangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A))] :
      ( member(A,B2,range(B,A,R))
     => ~ ! [A5: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),B2),R) ) ).

% RangeE
tff(fact_7879_Range__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,R: set(product_prod(B,A))] :
      ( member(A,A3,range(B,A,R))
    <=> ? [Y3: B] : member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Y3),A3),R) ) ).

% Range_iff
tff(fact_7880_disjnt__ge__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y6: set(A),X6: set(A)] :
          ( finite_finite(A,Y6)
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),Y6)),X4) )
           => disjnt(A,X6,Y6) ) ) ) ).

% disjnt_ge_max
tff(fact_7881_Rangep__Range__eq,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),X5: A] :
      ( aa(A,$o,rangep(B,A,aTP_Lamp_age(set(product_prod(B,A)),fun(B,fun(A,$o)),R)),X5)
    <=> member(A,X5,range(B,A,R)) ) ).

% Rangep_Range_eq
tff(fact_7882_Range__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : range(A,B,X5) = aa(fun(B,$o),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),X5))) ).

% Range_def
tff(fact_7883_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_li(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int_def
tff(fact_7884_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_lg(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int_def
tff(fact_7885_add_Ogroup__axioms,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => group(A,plus_plus(A),zero_zero(A),uminus_uminus(A)) ) ).

% add.group_axioms
tff(fact_7886_sub_Oabs__eq,axiom,
    ! [Xaa: num,Xb: num] : code_sub(Xaa,Xb) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Xaa)),aa(num,int,numeral_numeral(int),Xb))) ).

% sub.abs_eq
tff(fact_7887_Code__Numeral_Osub__code_I1_J,axiom,
    code_sub(one2,one2) = zero_zero(code_integer) ).

% Code_Numeral.sub_code(1)
tff(fact_7888_group_Oinverse__distrib__swap,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A,B2: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F2,A3),B2)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,B2)),aa(A,A,Inverse,A3)) ) ) ).

% group.inverse_distrib_swap
tff(fact_7889_group_Ogroup__left__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F2,Z),A3) = A3 ) ) ).

% group.group_left_neutral
tff(fact_7890_group_Oinverse__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,Inverse,Z) = Z ) ) ).

% group.inverse_neutral
tff(fact_7891_group_Oinverse__inverse,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,Inverse,A3)) = A3 ) ) ).

% group.inverse_inverse
tff(fact_7892_group_Oinverse__unique,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A,B2: A] :
      ( group(A,F2,Z,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = Z )
       => ( aa(A,A,Inverse,A3) = B2 ) ) ) ).

% group.inverse_unique
tff(fact_7893_group_Oright__inverse,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,Inverse,A3)) = Z ) ) ).

% group.right_inverse
tff(fact_7894_group_Oright__cancel,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),B2: A,A3: A,C2: A] :
      ( group(A,F2,Z,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,B2),A3) = aa(A,A,aa(A,fun(A,A),F2,C2),A3) )
      <=> ( B2 = C2 ) ) ) ).

% group.right_cancel
tff(fact_7895_group_Oleft__inverse,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A3)),A3) = Z ) ) ).

% group.left_inverse
tff(fact_7896_group_Oleft__cancel,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A,B2: A,C2: A] :
      ( group(A,F2,Z,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = aa(A,A,aa(A,fun(A,A),F2,A3),C2) )
      <=> ( B2 = C2 ) ) ) ).

% group.left_cancel
tff(fact_7897_sub_Orep__eq,axiom,
    ! [Xb: num,Xaa: num] : code_int_of_integer(code_sub(Xb,Xaa)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Xb)),aa(num,int,numeral_numeral(int),Xaa)) ).

% sub.rep_eq
tff(fact_7898_Code__Numeral_Osub__code_I9_J,axiom,
    ! [Mb: num,Nb: num] : code_sub(aa(num,num,bit0,Mb),aa(num,num,bit1,Nb)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),code_dup(code_sub(Mb,Nb))),one_one(code_integer)) ).

% Code_Numeral.sub_code(9)
tff(fact_7899_Code__Numeral_Osub__code_I8_J,axiom,
    ! [Mb: num,Nb: num] : code_sub(aa(num,num,bit1,Mb),aa(num,num,bit0,Nb)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_dup(code_sub(Mb,Nb))),one_one(code_integer)) ).

% Code_Numeral.sub_code(8)
tff(fact_7900_Code__Numeral_Osub__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : code_sub(aa(num,num,bit0,Mb),aa(num,num,bit0,Nb)) = code_dup(code_sub(Mb,Nb)) ).

% Code_Numeral.sub_code(6)
tff(fact_7901_Code__Numeral_Osub__code_I7_J,axiom,
    ! [Mb: num,Nb: num] : code_sub(aa(num,num,bit1,Mb),aa(num,num,bit1,Nb)) = code_dup(code_sub(Mb,Nb)) ).

% Code_Numeral.sub_code(7)
tff(fact_7902_Code__Numeral_Osub__code_I4_J,axiom,
    ! [Nb: num] : code_sub(one2,aa(num,num,bit0,Nb)) = code_Neg(bitM(Nb)) ).

% Code_Numeral.sub_code(4)
tff(fact_7903_Code__Numeral_Osub__code_I2_J,axiom,
    ! [Mb: num] : code_sub(aa(num,num,bit0,Mb),one2) = code_Pos(bitM(Mb)) ).

% Code_Numeral.sub_code(2)
tff(fact_7904_Code__Numeral_Odup__code_I3_J,axiom,
    ! [Nb: num] : code_dup(code_Neg(Nb)) = code_Neg(aa(num,num,bit0,Nb)) ).

% Code_Numeral.dup_code(3)
tff(fact_7905_Code__Numeral_Odup__code_I2_J,axiom,
    ! [Nb: num] : code_dup(code_Pos(Nb)) = code_Pos(aa(num,num,bit0,Nb)) ).

% Code_Numeral.dup_code(2)
tff(fact_7906_less__integer__code_I3_J,axiom,
    ! [L: num] : ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),code_Neg(L)) ).

% less_integer_code(3)
tff(fact_7907_less__integer__code_I7_J,axiom,
    ! [K: num] : aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_Neg(K)),zero_zero(code_integer)) ).

% less_integer_code(7)
tff(fact_7908_less__integer__code_I2_J,axiom,
    ! [L: num] : aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),code_Pos(L)) ).

% less_integer_code(2)
tff(fact_7909_less__integer__code_I4_J,axiom,
    ! [K: num] : ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_Pos(K)),zero_zero(code_integer)) ).

% less_integer_code(4)
tff(fact_7910_less__integer__code_I6_J,axiom,
    ! [K: num,L: num] : ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_Pos(K)),code_Neg(L)) ).

% less_integer_code(6)
tff(fact_7911_less__integer__code_I8_J,axiom,
    ! [K: num,L: num] : aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_Neg(K)),code_Pos(L)) ).

% less_integer_code(8)
tff(fact_7912_less__integer__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_Pos(K)),code_Pos(L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),K),L) ) ).

% less_integer_code(5)
tff(fact_7913_less__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_Neg(K)),code_Neg(L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),L),K) ) ).

% less_integer_code(9)
tff(fact_7914_plus__integer__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_Neg(Mb)),code_Neg(Nb)) = code_Neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ).

% plus_integer_code(6)
tff(fact_7915_Pos__fold_I3_J,axiom,
    ! [K: num] : aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,K)) = code_Pos(aa(num,num,bit1,K)) ).

% Pos_fold(3)
tff(fact_7916_Pos__fold_I2_J,axiom,
    ! [K: num] : aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,K)) = code_Pos(aa(num,num,bit0,K)) ).

% Pos_fold(2)
tff(fact_7917_Pos__fold_I1_J,axiom,
    aa(num,code_integer,numeral_numeral(code_integer),one2) = code_Pos(one2) ).

% Pos_fold(1)
tff(fact_7918_times__integer__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Neg(Mb)),code_Neg(Nb)) = code_Pos(aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb)) ).

% times_integer_code(6)
tff(fact_7919_times__integer__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Neg(Mb)),code_Pos(Nb)) = code_Neg(aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb)) ).

% times_integer_code(5)
tff(fact_7920_times__integer__code_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Pos(Mb)),code_Neg(Nb)) = code_Neg(aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb)) ).

% times_integer_code(4)
tff(fact_7921_one__integer__code,axiom,
    one_one(code_integer) = code_Pos(one2) ).

% one_integer_code
tff(fact_7922_plus__integer__code_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_Pos(Mb)),code_Pos(Nb)) = code_Pos(aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ).

% plus_integer_code(3)
tff(fact_7923_times__integer__code_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Pos(Mb)),code_Pos(Nb)) = code_Pos(aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb)) ).

% times_integer_code(3)
tff(fact_7924_minus__integer__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),code_Neg(Mb)),code_Pos(Nb)) = code_Neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ).

% minus_integer_code(5)
tff(fact_7925_minus__integer__code_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),code_Pos(Mb)),code_Neg(Nb)) = code_Pos(aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ).

% minus_integer_code(4)
tff(fact_7926_less__eq__integer__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),code_Pos(K)),code_Pos(L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),K),L) ) ).

% less_eq_integer_code(5)
tff(fact_7927_less__eq__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),code_Neg(K)),code_Neg(L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),L),K) ) ).

% less_eq_integer_code(9)
tff(fact_7928_Code__Numeral_Osub__code_I3_J,axiom,
    ! [Mb: num] : code_sub(aa(num,num,bit1,Mb),one2) = code_Pos(aa(num,num,bit0,Mb)) ).

% Code_Numeral.sub_code(3)
tff(fact_7929_Code__Numeral_Osub__code_I5_J,axiom,
    ! [Nb: num] : code_sub(one2,aa(num,num,bit1,Nb)) = code_Neg(aa(num,num,bit0,Nb)) ).

% Code_Numeral.sub_code(5)
tff(fact_7930_map__add__map__of__foldr,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),Ps: list(product_prod(A,B))] : map_add(A,B,Mb,map_of(A,B,Ps)) = foldr(product_prod(A,B),fun(A,option(B)),aa(fun(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),fun(product_prod(A,B),fun(fun(A,option(B)),fun(A,option(B)))),product_case_prod(A,B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_ahj(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))))),Ps,Mb) ).

% map_add_map_of_foldr
tff(fact_7931_wo__rel_OisMinim__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( bNF_We4791949203932849705sMinim(A,R,A4,B2)
      <=> ( member(A,B2,A4)
          & ! [X: A] :
              ( member(A,X,A4)
             => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),X),R) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_7932_map__add__find__right,axiom,
    ! [B: $tType,A: $tType,Nb: fun(B,option(A)),K: B,Xx: A,Mb: fun(B,option(A))] :
      ( ( aa(B,option(A),Nb,K) = aa(A,option(A),some(A),Xx) )
     => ( aa(B,option(A),map_add(B,A,Mb,Nb),K) = aa(A,option(A),some(A),Xx) ) ) ).

% map_add_find_right
tff(fact_7933_map__add__None,axiom,
    ! [B: $tType,A: $tType,Mb: fun(B,option(A)),Nb: fun(B,option(A)),K: B] :
      ( ( aa(B,option(A),map_add(B,A,Mb,Nb),K) = none(A) )
    <=> ( ( aa(B,option(A),Nb,K) = none(A) )
        & ( aa(B,option(A),Mb,K) = none(A) ) ) ) ).

% map_add_None
tff(fact_7934_map__add__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),G: fun(A,option(B))] :
      ( ! [X: A] : aa(A,option(B),map_add(A,B,F2,G),X) = none(B)
    <=> ( ! [X: A] : aa(A,option(B),F2,X) = none(B)
        & ! [X: A] : aa(A,option(B),G,X) = none(B) ) ) ).

% map_add_eq_empty_iff
tff(fact_7935_empty__map__add,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B))] : map_add(A,B,aTP_Lamp_on(A,option(B)),Mb) = Mb ).

% empty_map_add
tff(fact_7936_map__add__empty,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B))] : map_add(A,B,Mb,aTP_Lamp_on(A,option(B))) = Mb ).

% map_add_empty
tff(fact_7937_empty__eq__map__add__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( aTP_Lamp_on(A,option(B)) = map_add(A,B,F2,G) )
    <=> ( ! [X: A] : aa(A,option(B),F2,X) = none(B)
        & ! [X: A] : aa(A,option(B),G,X) = none(B) ) ) ).

% empty_eq_map_add_iff
tff(fact_7938_map__add__upd,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),G: fun(A,option(B)),Xb: A,Y: B] : map_add(A,B,F2,fun_upd(A,option(B),G,Xb,aa(B,option(B),some(B),Y))) = fun_upd(A,option(B),map_add(A,B,F2,G),Xb,aa(B,option(B),some(B),Y)) ).

% map_add_upd
tff(fact_7939_map__add__Some__iff,axiom,
    ! [B: $tType,A: $tType,Mb: fun(B,option(A)),Nb: fun(B,option(A)),K: B,Xb: A] :
      ( ( aa(B,option(A),map_add(B,A,Mb,Nb),K) = aa(A,option(A),some(A),Xb) )
    <=> ( ( aa(B,option(A),Nb,K) = aa(A,option(A),some(A),Xb) )
        | ( ( aa(B,option(A),Nb,K) = none(A) )
          & ( aa(B,option(A),Mb,K) = aa(A,option(A),some(A),Xb) ) ) ) ) ).

% map_add_Some_iff
tff(fact_7940_map__add__SomeD,axiom,
    ! [B: $tType,A: $tType,Mb: fun(B,option(A)),Nb: fun(B,option(A)),K: B,Xb: A] :
      ( ( aa(B,option(A),map_add(B,A,Mb,Nb),K) = aa(A,option(A),some(A),Xb) )
     => ( ( aa(B,option(A),Nb,K) = aa(A,option(A),some(A),Xb) )
        | ( ( aa(B,option(A),Nb,K) = none(A) )
          & ( aa(B,option(A),Mb,K) = aa(A,option(A),some(A),Xb) ) ) ) ) ).

% map_add_SomeD
tff(fact_7941_map__add__def,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X5: A] : aa(A,option(B),map_add(A,B,M1,M22),X5) = aa(option(B),option(B),aa(fun(B,option(B)),fun(option(B),option(B)),aa(option(B),fun(fun(B,option(B)),fun(option(B),option(B))),case_option(option(B),B),aa(A,option(B),M1,X5)),some(B)),aa(A,option(B),M22,X5)) ).

% map_add_def
tff(fact_7942_map__add__upd__left,axiom,
    ! [A: $tType,B: $tType,Mb: A,E22: fun(A,option(B)),E1: fun(A,option(B)),U1: B] :
      ( ~ member(A,Mb,dom(A,B,E22))
     => ( map_add(A,B,fun_upd(A,option(B),E1,Mb,aa(B,option(B),some(B),U1)),E22) = fun_upd(A,option(B),map_add(A,B,E1,E22),Mb,aa(B,option(B),some(B),U1)) ) ) ).

% map_add_upd_left
tff(fact_7943_subset__maxchain__max,axiom,
    ! [A: $tType,A4: set(set(A)),C5: set(set(A)),X6: set(A)] :
      ( pred_maxchain(set(A),A4,ord_less(set(A)),C5)
     => ( member(set(A),X6,A4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)),X6)
         => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5) = X6 ) ) ) ) ).

% subset_maxchain_max
tff(fact_7944_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A)] : aa(fun(fun(A,$o),fun(A,$o)),$o,order_mono(fun(A,$o),fun(A,$o)),aTP_Lamp_ahk(fun(A,A),fun(fun(A,$o),fun(A,$o)),F2)) ) ).

% iteratesp.mono
tff(fact_7945_subset_OHausdorff,axiom,
    ! [A: $tType,A4: set(set(A))] :
    ? [X_12: set(set(A))] : pred_maxchain(set(A),A4,ord_less(set(A)),X_12) ).

% subset.Hausdorff
tff(fact_7946_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xb: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_7947_ccpo__Sup__upper,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A4: set(A),Xb: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
         => ( member(A,Xb,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ).

% ccpo_Sup_upper
tff(fact_7948_ccpo__Sup__least,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A4: set(A),Z: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
         => ( ! [X4: A] :
                ( member(A,X4,A4)
               => aa(A,$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),A4)),Z) ) ) ) ).

% ccpo_Sup_least
tff(fact_7949_subset_Omaxchain__def,axiom,
    ! [A: $tType,A4: set(set(A)),C5: set(set(A))] :
      ( pred_maxchain(set(A),A4,ord_less(set(A)),C5)
    <=> ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C5)
        & ~ ? [S8: set(set(A))] :
              ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),S8)
              & aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less(set(set(A))),C5),S8) ) ) ) ).

% subset.maxchain_def
tff(fact_7950_subset_Omaxchain__imp__chain,axiom,
    ! [A: $tType,A4: set(set(A)),C5: set(set(A))] :
      ( pred_maxchain(set(A),A4,ord_less(set(A)),C5)
     => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C5) ) ).

% subset.maxchain_imp_chain
tff(fact_7951_pred__on_Omaxchain__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C5: set(A)] :
      ( pred_maxchain(A,A4,P,C5)
    <=> ( aa(set(A),$o,pred_chain(A,A4,P),C5)
        & ~ ? [S8: set(A)] :
              ( aa(set(A),$o,pred_chain(A,A4,P),S8)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),C5),S8) ) ) ) ).

% pred_on.maxchain_def
tff(fact_7952_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A4: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
         => ( finite_finite(A,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),A4),A4) ) ) ) ) ).

% in_chain_finite
tff(fact_7953_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X5: fun(A,A)] : comple7512665784863727008ratesp(A,X5) = complete_lattice_lfp(fun(A,$o),aTP_Lamp_ahk(fun(A,A),fun(fun(A,$o),fun(A,$o)),X5)) ) ).

% iteratesp_def
tff(fact_7954_iteratesp_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M6: set(A),F2: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M6)
         => ( ! [X4: A] :
                ( member(A,X4,M6)
               => aa(A,$o,comple7512665784863727008ratesp(A,F2),X4) )
           => aa(A,$o,comple7512665784863727008ratesp(A,F2),aa(set(A),A,complete_Sup_Sup(A),M6)) ) ) ) ).

% iteratesp.Sup
tff(fact_7955_iteratesp_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),A3: A] :
          ( aa(A,$o,comple7512665784863727008ratesp(A,F2),A3)
         => ( ! [X4: A] :
                ( ( A3 = aa(A,A,F2,X4) )
               => ~ aa(A,$o,comple7512665784863727008ratesp(A,F2),X4) )
           => ~ ! [M7: set(A)] :
                  ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M7) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M7)
                   => ~ ! [X5: A] :
                          ( member(A,X5,M7)
                         => aa(A,$o,comple7512665784863727008ratesp(A,F2),X5) ) ) ) ) ) ) ).

% iteratesp.cases
tff(fact_7956_iteratesp_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),A3: A] :
          ( aa(A,$o,comple7512665784863727008ratesp(A,F2),A3)
        <=> ( ? [X: A] :
                ( ( A3 = aa(A,A,F2,X) )
                & aa(A,$o,comple7512665784863727008ratesp(A,F2),X) )
            | ? [M8: set(A)] :
                ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X: A] :
                    ( member(A,X,M8)
                   => aa(A,$o,comple7512665784863727008ratesp(A,F2),X) ) ) ) ) ) ).

% iteratesp.simps
tff(fact_7957_subset_Onot__maxchain__Some,axiom,
    ! [A: $tType,A4: set(set(A)),C5: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C5)
     => ( ~ pred_maxchain(set(A),A4,ord_less(set(A)),C5)
       => ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_ahl(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),A4),C5)))
          & aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less(set(set(A))),C5),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_ahl(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),A4),C5))) ) ) ) ).

% subset.not_maxchain_Some
tff(fact_7958_admissible__chfin,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,$o)] :
          ( ! [S4: set(A)] :
              ( comple1602240252501008431_chain(A,ord_less_eq(A),S4)
             => finite_finite(A,S4) )
         => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P) ) ) ).

% admissible_chfin
tff(fact_7959_admissible__disj,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
         => ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),Q)
           => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) ) ) ) ).

% admissible_disj
tff(fact_7960_pred__on_Onot__maxchain__Some,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C5: set(A)] :
      ( aa(set(A),$o,pred_chain(A,A4,P),C5)
     => ( ~ pred_maxchain(A,A4,P,C5)
       => ( aa(set(A),$o,pred_chain(A,A4,P),fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_ahn(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),A4),P),C5)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),C5),fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_ahn(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),A4),P),C5))) ) ) ) ).

% pred_on.not_maxchain_Some
tff(fact_7961_subset_Osuc__def,axiom,
    ! [A: $tType,A4: set(set(A)),C5: set(set(A))] :
      pred_suc(set(A),A4,ord_less(set(A)),C5) = $ite(
        ( ~ aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C5)
        | pred_maxchain(set(A),A4,ord_less(set(A)),C5) ),
        C5,
        fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_ahl(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),A4),C5)) ) ).

% subset.suc_def
tff(fact_7962_pred__on_Osuc__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C5: set(A)] :
      pred_suc(A,A4,P,C5) = $ite(
        ( ~ aa(set(A),$o,pred_chain(A,A4,P),C5)
        | pred_maxchain(A,A4,P,C5) ),
        C5,
        fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_ahn(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),A4),P),C5)) ) ).

% pred_on.suc_def
tff(fact_7963_subset_Onot__chain__suc,axiom,
    ! [A: $tType,A4: set(set(A)),X6: set(set(A))] :
      ( ~ aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),X6)
     => ( pred_suc(set(A),A4,ord_less(set(A)),X6) = X6 ) ) ).

% subset.not_chain_suc
tff(fact_7964_subset_Omaxchain__suc,axiom,
    ! [A: $tType,A4: set(set(A)),X6: set(set(A))] :
      ( pred_maxchain(set(A),A4,ord_less(set(A)),X6)
     => ( pred_suc(set(A),A4,ord_less(set(A)),X6) = X6 ) ) ).

% subset.maxchain_suc
tff(fact_7965_subset_Osuc__in__carrier,axiom,
    ! [A: $tType,X6: set(set(A)),A4: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X6),A4)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pred_suc(set(A),A4,ord_less(set(A)),X6)),A4) ) ).

% subset.suc_in_carrier
tff(fact_7966_subset_Osuc__subset,axiom,
    ! [A: $tType,X6: set(set(A)),A4: set(set(A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X6),pred_suc(set(A),A4,ord_less(set(A)),X6)) ).

% subset.suc_subset
tff(fact_7967_subset_Osubset__suc,axiom,
    ! [A: $tType,X6: set(set(A)),Y6: set(set(A)),A4: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X6),Y6)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X6),pred_suc(set(A),A4,ord_less(set(A)),Y6)) ) ).

% subset.subset_suc
tff(fact_7968_subset_Ochain__suc,axiom,
    ! [A: $tType,A4: set(set(A)),X6: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),X6)
     => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),pred_suc(set(A),A4,ord_less(set(A)),X6)) ) ).

% subset.chain_suc
tff(fact_7969_subset_Ochain__sucD,axiom,
    ! [A: $tType,A4: set(set(A)),X6: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),X6)
     => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pred_suc(set(A),A4,ord_less(set(A)),X6)),A4)
        & aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),pred_suc(set(A),A4,ord_less(set(A)),X6)) ) ) ).

% subset.chain_sucD
tff(fact_7970_subset_Osuc__not__equals,axiom,
    ! [A: $tType,A4: set(set(A)),C5: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C5)
     => ( ~ pred_maxchain(set(A),A4,ord_less(set(A)),C5)
       => ( pred_suc(set(A),A4,ord_less(set(A)),C5) != C5 ) ) ) ).

% subset.suc_not_equals
tff(fact_7971_and_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.monoid_axioms
tff(fact_7972_splice__replicate,axiom,
    ! [A: $tType,Mb: nat,Xb: A,Nb: nat] : splice(A,replicate(A,Mb,Xb),replicate(A,Nb,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb),Xb) ).

% splice_replicate
tff(fact_7973_length__splice,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),splice(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_splice
tff(fact_7974_max__nat_Omonoid__axioms,axiom,
    monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.monoid_axioms
tff(fact_7975_xor_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.monoid_axioms
tff(fact_7976_or_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.monoid_axioms
tff(fact_7977_add_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.monoid_axioms
tff(fact_7978_monoid_Oright__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A3: A] :
      ( monoid(A,F2,Z)
     => ( aa(A,A,aa(A,fun(A,A),F2,A3),Z) = A3 ) ) ).

% monoid.right_neutral
tff(fact_7979_monoid_Oleft__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A3: A] :
      ( monoid(A,F2,Z)
     => ( aa(A,A,aa(A,fun(A,A),F2,Z),A3) = A3 ) ) ).

% monoid.left_neutral
tff(fact_7980_mult_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => monoid(A,times_times(A),one_one(A)) ) ).

% mult.monoid_axioms
tff(fact_7981_VEBT__internal_Ooption__shift_Opelims,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 )
     => ( accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A),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))),Xb),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)),Xaa),Xba)))
       => ( ( ( Xaa = none(A) )
           => ( ( Y = none(A) )
             => ~ accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A),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))),Xb),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)),Xba))) ) )
         => ( ! [V3: A] :
                ( ( Xaa = aa(A,option(A),some(A),V3) )
               => ( ( Xba = none(A) )
                 => ( ( Y = none(A) )
                   => ~ accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A),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))),Xb),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)))) ) ) )
           => ~ ! [A5: A] :
                  ( ( Xaa = aa(A,option(A),some(A),A5) )
                 => ! [B3: A] :
                      ( ( Xba = aa(A,option(A),some(A),B3) )
                     => ( ( Y = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Xb,A5),B3)) )
                       => ~ accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A),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))),Xb),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),A5)),aa(A,option(A),some(A),B3)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7982_dim__le__card_H,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( finite_finite(A,S)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),real_Vector_dim(A,S)),aa(set(A),nat,finite_card(A),S)) ) ) ).

% dim_le_card'
tff(fact_7983_span__card__ge__dim,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A),V2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),V2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V2),real_Vector_span(A,B4))
           => ( finite_finite(A,B4)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),real_Vector_dim(A,V2)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ) ).

% span_card_ge_dim
tff(fact_7984_dim__le__card,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V2: set(A),W3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V2),real_Vector_span(A,W3))
         => ( finite_finite(A,W3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),real_Vector_dim(A,V2)),aa(set(A),nat,finite_card(A),W3)) ) ) ) ).

% dim_le_card
tff(fact_7985_span__induct__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,S3: set(A),H: fun(A,$o)] :
          ( member(A,Xb,real_Vector_span(A,S3))
         => ( aa(A,$o,H,zero_zero(A))
           => ( ! [C3: real,X4: A,Y4: A] :
                  ( member(A,X4,S3)
                 => ( aa(A,$o,H,Y4)
                   => aa(A,$o,H,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),X4)),Y4)) ) )
             => aa(A,$o,H,Xb) ) ) ) ) ).

% span_induct_alt
tff(fact_7986_span__add__eq2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Y: A,S3: set(A),Xb: A] :
          ( member(A,Y,real_Vector_span(A,S3))
         => ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),real_Vector_span(A,S3))
          <=> member(A,Xb,real_Vector_span(A,S3)) ) ) ) ).

% span_add_eq2
tff(fact_7987_span__add__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,S3: set(A),Y: A] :
          ( member(A,Xb,real_Vector_span(A,S3))
         => ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),real_Vector_span(A,S3))
          <=> member(A,Y,real_Vector_span(A,S3)) ) ) ) ).

% span_add_eq
tff(fact_7988_span__add,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,S3: set(A),Y: A] :
          ( member(A,Xb,real_Vector_span(A,S3))
         => ( member(A,Y,real_Vector_span(A,S3))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),real_Vector_span(A,S3)) ) ) ) ).

% span_add
tff(fact_7989_span__Un,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A),T5: set(A)] : real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S3),T5)) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aho(set(A),fun(set(A),fun(A,$o)),S3),T5)) ) ).

% span_Un
tff(fact_7990_independent__span__bound,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [T5: set(A),S3: set(A)] :
          ( finite_finite(A,T5)
         => ( ~ real_V358717886546972837endent(A,S3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),real_Vector_span(A,T5))
             => ( finite_finite(A,S3)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),S3)),aa(set(A),nat,finite_card(A),T5)) ) ) ) ) ) ).

% independent_span_bound
tff(fact_7991_Domain__insert,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] : domain(A,B,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),A3),B2)),R)) = aa(set(A),set(A),insert(A,A3),domain(A,B,R)) ).

% Domain_insert
tff(fact_7992_Not__Domain__rtrancl,axiom,
    ! [A: $tType,Xb: A,R2: set(product_prod(A,A)),Y: A] :
      ( ~ member(A,Xb,domain(A,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))
      <=> ( Xb = Y ) ) ) ).

% Not_Domain_rtrancl
tff(fact_7993_Domain_Ocases,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( member(A,A3,domain(A,B,R))
     => ~ ! [B3: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R) ) ).

% Domain.cases
tff(fact_7994_Domain_Osimps,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( member(A,A3,domain(A,B,R))
    <=> ? [A6: A,B5: B] :
          ( ( A3 = A6 )
          & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5),R) ) ) ).

% Domain.simps
tff(fact_7995_Domain_ODomainI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(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),A3),B2),R)
     => member(A,A3,domain(A,B,R)) ) ).

% Domain.DomainI
tff(fact_7996_DomainE,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( member(A,A3,domain(A,B,R))
     => ~ ! [B3: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R) ) ).

% DomainE
tff(fact_7997_Domain__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( member(A,A3,domain(A,B,R))
    <=> ? [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),A3),Y3),R) ) ).

% Domain_iff
tff(fact_7998_Domain__unfold,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : domain(A,B,R) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ahp(set(product_prod(A,B)),fun(A,$o),R)) ).

% Domain_unfold
tff(fact_7999_representation__add,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A,U: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( member(A,V,real_Vector_span(A,Basis))
           => ( member(A,U,real_Vector_span(A,Basis))
             => ! [X5: A] : real_V7696804695334737415tation(A,Basis,aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V),X5) = aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7696804695334737415tation(A,Basis,U,X5)),real_V7696804695334737415tation(A,Basis,V,X5)) ) ) ) ) ).

% representation_add
tff(fact_8000_ndepth__Push__Node__aux,axiom,
    ! [A: $tType,I: nat,F2: fun(nat,sum_sum(A,nat)),K: nat] :
      ( ( case_nat(sum_sum(A,nat),aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,I)),F2,K) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_ahq(fun(nat,sum_sum(A,nat)),fun(nat,$o),F2)))),K) ) ).

% ndepth_Push_Node_aux
tff(fact_8001_sum_Osize__gen_I2_J,axiom,
    ! [A: $tType,B: $tType,Xaa: fun(A,nat),Xb: fun(B,nat),X22: B] : basic_BNF_size_sum(A,B,Xaa,Xb,aa(B,sum_sum(A,B),sum_Inr(B,A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,Xb,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% sum.size_gen(2)
tff(fact_8002_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Y: A,Zs: list(A)] :
          arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) = $let(
            m2: A,
            m2:= arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,m2)),Xb,m2) ) ) ).

% arg_min_list.simps(2)
tff(fact_8003_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),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) )
               => ( Y != X4 ) )
           => ( ! [X4: A,Y4: A,Zs2: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)) )
                 => ( Y != $let(
                        m2: A,
                        m2:= arg_min_list(A,B,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),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_8004_sum_Osize__gen_I1_J,axiom,
    ! [B: $tType,A: $tType,Xaa: fun(A,nat),Xb: fun(B,nat),X1: A] : basic_BNF_size_sum(A,B,Xaa,Xb,aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xaa,X1)),aa(nat,nat,suc,zero_zero(nat))) ).

% sum.size_gen(1)
tff(fact_8005_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),undefined(A)),aTP_Lamp_mf(A,A)),Option) ).

% option.the_def
tff(fact_8006_sum__decode__def,axiom,
    ! [Nb: nat] :
      nat_sum_decode(Nb) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb),aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% sum_decode_def
tff(fact_8007_prod_OPlus,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),B4: set(B),G: fun(sum_sum(A,B),C)] :
          ( finite_finite(A,A4)
         => ( finite_finite(B,B4)
           => ( aa(set(sum_sum(A,B)),C,aa(fun(sum_sum(A,B),C),fun(set(sum_sum(A,B)),C),groups7121269368397514597t_prod(sum_sum(A,B),C),G),sum_Plus(A,B,A4,B4)) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,sum_sum(A,B)),fun(A,C),comp(sum_sum(A,B),C,A,G),sum_Inl(A,B))),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(B,sum_sum(A,B)),fun(B,C),comp(sum_sum(A,B),C,B,G),sum_Inr(B,A))),B4)) ) ) ) ) ).

% prod.Plus
tff(fact_8008_card__Plus,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B4: set(B)] :
      ( finite_finite(A,A4)
     => ( finite_finite(B,B4)
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A4,B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ).

% card_Plus
tff(fact_8009_card__Plus__conv__if,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B4: set(B)] :
      aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A4,B4)) = $ite(
        ( finite_finite(A,A4)
        & finite_finite(B,B4) ),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B4)),
        zero_zero(nat) ) ).

% card_Plus_conv_if
tff(fact_8010_sum_OPlus,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(A),B4: set(B),G: fun(sum_sum(A,B),C)] :
          ( finite_finite(A,A4)
         => ( finite_finite(B,B4)
           => ( aa(set(sum_sum(A,B)),C,groups7311177749621191930dd_sum(sum_sum(A,B),C,G),sum_Plus(A,B,A4,B4)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(A,sum_sum(A,B)),fun(A,C),comp(sum_sum(A,B),C,A,G),sum_Inl(A,B))),A4)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(B,sum_sum(A,B)),fun(B,C),comp(sum_sum(A,B),C,B,G),sum_Inr(B,A))),B4)) ) ) ) ) ).

% sum.Plus
tff(fact_8011_nth__item_Opinduct,axiom,
    ! [A0: nat,P: fun(nat,$o)] :
      ( accp(nat,nth_item_rel,A0)
     => ( ( accp(nat,nth_item_rel,zero_zero(nat))
         => aa(nat,$o,P,zero_zero(nat)) )
       => ( ! [N: nat] :
              ( accp(nat,nth_item_rel,aa(nat,nat,suc,N))
             => ( ! [A9: nat,Aa3: nat] :
                    ( ( nat_sum_decode(N) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A9) )
                   => ( ( nat_sum_decode(A9) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),Aa3) )
                     => aa(nat,$o,P,Aa3) ) )
               => ( ! [A9: nat,B10: nat] :
                      ( ( nat_sum_decode(N) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A9) )
                     => ( ( nat_sum_decode(A9) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B10) )
                       => aa(nat,$o,P,B10) ) )
                 => ( ! [B10: nat,Ba2: nat,X5: nat,Y5: nat] :
                        ( ( nat_sum_decode(N) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B10) )
                       => ( ( nat_sum_decode(B10) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba2) )
                         => ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X5),Y5) = nat_prod_decode(Ba2) )
                           => aa(nat,$o,P,X5) ) ) )
                   => ( ! [B10: nat,Ba2: nat,X5: nat,Y5: nat] :
                          ( ( nat_sum_decode(N) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B10) )
                         => ( ( nat_sum_decode(B10) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba2) )
                           => ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X5),Y5) = nat_prod_decode(Ba2) )
                             => aa(nat,$o,P,Y5) ) ) )
                     => aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ) )
         => aa(nat,$o,P,A0) ) ) ) ).

% nth_item.pinduct
tff(fact_8012_le__sum__encode__Inl,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),nat_sum_encode(aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),Y))) ) ).

% le_sum_encode_Inl
tff(fact_8013_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: fun(A,B),Xaa: list(A),Y: A] :
          ( ( arg_min_list(A,B,Xb,Xaa) = Y )
         => ( accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),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)),Xb),Xaa))
           => ( ! [X4: A] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) )
                 => ( ( Y = X4 )
                   => ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),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)),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)))) ) )
             => ( ! [X4: A,Y4: A,Zs2: list(A)] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)) )
                   => ( ( Y = $let(
                            m2: A,
                            m2:= arg_min_list(A,B,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),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) ) )
                     => ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),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)),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)))) ) )
               => ~ ( ( Xaa = nil(A) )
                   => ( ( Y = undefined(A) )
                     => ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),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)),Xb),nil(A))) ) ) ) ) ) ) ) ).

% arg_min_list.pelims
tff(fact_8014_le__sum__encode__Inr,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),nat_sum_encode(aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Y))) ) ).

% le_sum_encode_Inr
tff(fact_8015_sum__encode__def,axiom,
    ! [Xb: sum_sum(nat,nat)] : nat_sum_encode(Xb) = sum_case_sum(nat,nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aTP_Lamp_ahr(nat,nat),Xb) ).

% sum_encode_def
tff(fact_8016_infinity__enat__def,axiom,
    extend4730790105801354508finity(extended_enat) = aa(option(nat),extended_enat,extended_Abs_enat,none(nat)) ).

% infinity_enat_def
tff(fact_8017_Abs__enat__cases,axiom,
    ! [Xb: extended_enat] :
      ~ ! [Y4: option(nat)] :
          ( ( Xb = aa(option(nat),extended_enat,extended_Abs_enat,Y4) )
         => ~ member(option(nat),Y4,top_top(set(option(nat)))) ) ).

% Abs_enat_cases
tff(fact_8018_Abs__enat__induct,axiom,
    ! [P: fun(extended_enat,$o),Xb: extended_enat] :
      ( ! [Y4: option(nat)] :
          ( member(option(nat),Y4,top_top(set(option(nat))))
         => aa(extended_enat,$o,P,aa(option(nat),extended_enat,extended_Abs_enat,Y4)) )
     => aa(extended_enat,$o,P,Xb) ) ).

% Abs_enat_induct
tff(fact_8019_Abs__enat__inject,axiom,
    ! [Xb: option(nat),Y: option(nat)] :
      ( member(option(nat),Xb,top_top(set(option(nat))))
     => ( member(option(nat),Y,top_top(set(option(nat))))
       => ( ( aa(option(nat),extended_enat,extended_Abs_enat,Xb) = aa(option(nat),extended_enat,extended_Abs_enat,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% Abs_enat_inject
tff(fact_8020_enat__def,axiom,
    ! [Nb: nat] : extended_enat2(Nb) = aa(option(nat),extended_enat,extended_Abs_enat,aa(nat,option(nat),some(nat),Nb)) ).

% enat_def
tff(fact_8021_iterates_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: A,F2: fun(A,A)] :
          ( member(A,A3,comple6359979572994053840erates(A,F2))
        <=> ( ? [X: A] :
                ( ( A3 = aa(A,A,F2,X) )
                & member(A,X,comple6359979572994053840erates(A,F2)) )
            | ? [M8: set(A)] :
                ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X: A] :
                    ( member(A,X,M8)
                   => member(A,X,comple6359979572994053840erates(A,F2)) ) ) ) ) ) ).

% iterates.simps
tff(fact_8022_iterates_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M6: set(A),F2: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M6)
         => ( ! [X4: A] :
                ( member(A,X4,M6)
               => member(A,X4,comple6359979572994053840erates(A,F2)) )
           => member(A,aa(set(A),A,complete_Sup_Sup(A),M6),comple6359979572994053840erates(A,F2)) ) ) ) ).

% iterates.Sup
tff(fact_8023_iterates_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: A,F2: fun(A,A)] :
          ( member(A,A3,comple6359979572994053840erates(A,F2))
         => ( ! [X4: A] :
                ( ( A3 = aa(A,A,F2,X4) )
               => ~ member(A,X4,comple6359979572994053840erates(A,F2)) )
           => ~ ! [M7: set(A)] :
                  ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M7) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M7)
                   => ~ ! [X5: A] :
                          ( member(A,X5,M7)
                         => member(A,X5,comple6359979572994053840erates(A,F2)) ) ) ) ) ) ) ).

% iterates.cases
tff(fact_8024_Abs__enat__inverse,axiom,
    ! [Y: option(nat)] :
      ( member(option(nat),Y,top_top(set(option(nat))))
     => ( aa(extended_enat,option(nat),extended_Rep_enat,aa(option(nat),extended_enat,extended_Abs_enat,Y)) = Y ) ) ).

% Abs_enat_inverse
tff(fact_8025_chain__iterates,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A)] :
          ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F2)
         => comple1602240252501008431_chain(A,ord_less_eq(A),comple6359979572994053840erates(A,F2)) ) ) ).

% chain_iterates
tff(fact_8026_iterates__le__f,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xb: A,F2: fun(A,A)] :
          ( member(A,Xb,comple6359979572994053840erates(A,F2))
         => ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,F2,Xb)) ) ) ) ).

% iterates_le_f
tff(fact_8027_Rep__enat__induct,axiom,
    ! [Y: option(nat),P: fun(option(nat),$o)] :
      ( member(option(nat),Y,top_top(set(option(nat))))
     => ( ! [X4: extended_enat] : aa(option(nat),$o,P,aa(extended_enat,option(nat),extended_Rep_enat,X4))
       => aa(option(nat),$o,P,Y) ) ) ).

% Rep_enat_induct
tff(fact_8028_Rep__enat__cases,axiom,
    ! [Y: option(nat)] :
      ( member(option(nat),Y,top_top(set(option(nat))))
     => ~ ! [X4: extended_enat] : Y != aa(extended_enat,option(nat),extended_Rep_enat,X4) ) ).

% Rep_enat_cases
tff(fact_8029_Rep__enat,axiom,
    ! [Xb: extended_enat] : member(option(nat),aa(extended_enat,option(nat),extended_Rep_enat,Xb),top_top(set(option(nat)))) ).

% Rep_enat
tff(fact_8030_Rep__enat__inject,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ( aa(extended_enat,option(nat),extended_Rep_enat,Xb) = aa(extended_enat,option(nat),extended_Rep_enat,Y) )
    <=> ( Xb = Y ) ) ).

% Rep_enat_inject
tff(fact_8031_Rep__enat__inverse,axiom,
    ! [Xb: extended_enat] : aa(option(nat),extended_enat,extended_Abs_enat,aa(extended_enat,option(nat),extended_Rep_enat,Xb)) = Xb ).

% Rep_enat_inverse
tff(fact_8032_fixp__induct,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,$o),F2: fun(A,A)] :
          ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
         => ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F2)
           => ( aa(A,$o,P,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))))
             => ( ! [X4: A] :
                    ( aa(A,$o,P,X4)
                   => aa(A,$o,P,aa(A,A,F2,X4)) )
               => aa(A,$o,P,comple115746919287870866o_fixp(A,F2)) ) ) ) ) ) ).

% fixp_induct
tff(fact_8033_iterates__fixp,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A)] :
          ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F2)
         => member(A,comple115746919287870866o_fixp(A,F2),comple6359979572994053840erates(A,F2)) ) ) ).

% iterates_fixp
tff(fact_8034_fixp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),Z: A] :
          ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,Z)),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),comple115746919287870866o_fixp(A,F2)),Z) ) ) ) ).

% fixp_lowerbound
tff(fact_8035_fixp__unfold,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A)] :
          ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F2)
         => ( comple115746919287870866o_fixp(A,F2) = aa(A,A,F2,comple115746919287870866o_fixp(A,F2)) ) ) ) ).

% fixp_unfold
tff(fact_8036_type__definition__enat,axiom,
    type_definition(extended_enat,option(nat),extended_Rep_enat,extended_Abs_enat,top_top(set(option(nat)))) ).

% type_definition_enat
tff(fact_8037_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F2,linorder_sort_key(B,A,F2,Xs))) ) ).

% sorted_sort_key
tff(fact_8038_length__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),linorder_sort_key(A,B,F2,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ).

% length_sort
tff(fact_8039_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( linorder_sort_key(A,A,aTP_Lamp_ov(A,A),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_8040_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),linorder_sort_key(A,A,aTP_Lamp_ov(A,A),Xs)) ) ).

% sorted_sort
tff(fact_8041_map__conv__bind__option,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xb: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),Xb) = aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),Xb),aa(fun(B,A),fun(B,option(A)),comp(A,option(A),B,some(A)),F2)) ).

% map_conv_bind_option
tff(fact_8042_continuous__attains__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F2)
             => ? [X4: A] :
                  ( member(A,X4,S)
                  & ! [Xa: A] :
                      ( member(A,Xa,S)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Xa)) ) ) ) ) ) ) ).

% continuous_attains_inf
tff(fact_8043_bind__assoc,axiom,
    ! [C: $tType,A: $tType,B: $tType,Xb: option(C),F2: fun(C,option(B)),G: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(fun(C,option(B)),option(B),aa(option(C),fun(fun(C,option(B)),option(B)),bind(C,B),Xb),F2)),G) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),Xb),aa(fun(B,option(A)),fun(C,option(A)),aTP_Lamp_ahs(fun(C,option(B)),fun(fun(B,option(A)),fun(C,option(A))),F2),G)) ).

% bind_assoc
tff(fact_8044_bind__runit,axiom,
    ! [A: $tType,Xb: option(A)] : aa(fun(A,option(A)),option(A),aa(option(A),fun(fun(A,option(A)),option(A)),bind(A,A),Xb),some(A)) = Xb ).

% bind_runit
tff(fact_8045_bind__rzero,axiom,
    ! [B: $tType,A: $tType,Xb: option(B)] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),Xb),aTP_Lamp_nt(B,option(A))) = none(A) ).

% bind_rzero
tff(fact_8046_compact__attains__inf,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A)] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,S3)
                & ! [Xa: A] :
                    ( member(A,Xa,S3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa) ) ) ) ) ) ).

% compact_attains_inf
tff(fact_8047_compact__attains__sup,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A)] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,S3)
                & ! [Xa: A] :
                    ( member(A,Xa,S3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X4) ) ) ) ) ) ).

% compact_attains_sup
tff(fact_8048_bind__eq__None__conv,axiom,
    ! [B: $tType,A: $tType,A3: option(B),F2: fun(B,option(A))] :
      ( ( aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),A3),F2) = none(A) )
    <=> ( ( A3 = none(B) )
        | ( aa(B,option(A),F2,aa(option(B),B,the2(B),A3)) = none(A) ) ) ) ).

% bind_eq_None_conv
tff(fact_8049_bind__option__cong__code,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),Y: option(A),F2: fun(A,option(B))] :
      ( ( Xb = Y )
     => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Xb),F2) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y),F2) ) ) ).

% bind_option_cong_code
tff(fact_8050_bind_Obind__lzero,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),none(B)),F2) = none(A) ).

% bind.bind_lzero
tff(fact_8051_bind_Obind__lunit,axiom,
    ! [A: $tType,B: $tType,Xb: B,F2: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(B,option(B),some(B),Xb)),F2) = aa(B,option(A),F2,Xb) ).

% bind.bind_lunit
tff(fact_8052_Option_Obind__cong,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),Y: option(A),F2: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( Xb = Y )
     => ( ! [A5: A] :
            ( ( Y = aa(A,option(A),some(A),A5) )
           => ( aa(A,option(B),F2,A5) = aa(A,option(B),G,A5) ) )
       => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Xb),F2) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y),G) ) ) ) ).

% Option.bind_cong
tff(fact_8053_bind__eq__Some__conv,axiom,
    ! [B: $tType,A: $tType,F2: option(B),G: fun(B,option(A)),Xb: A] :
      ( ( aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),F2),G) = aa(A,option(A),some(A),Xb) )
    <=> ? [Y3: B] :
          ( ( F2 = aa(B,option(B),some(B),Y3) )
          & ( aa(B,option(A),G,Y3) = aa(A,option(A),some(A),Xb) ) ) ) ).

% bind_eq_Some_conv
tff(fact_8054_bind__split__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(option(A),$o),Mb: option(B),F2: fun(B,option(A))] :
      ( aa(option(A),$o,P,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),Mb),F2))
    <=> ~ ( ( ( Mb = none(B) )
            & ~ aa(option(A),$o,P,none(A)) )
          | ? [X: B] :
              ( ( Mb = aa(B,option(B),some(B),X) )
              & ~ aa(option(A),$o,P,aa(B,option(A),F2,X)) ) ) ) ).

% bind_split_asm
tff(fact_8055_bind__split,axiom,
    ! [A: $tType,B: $tType,P: fun(option(A),$o),Mb: option(B),F2: fun(B,option(A))] :
      ( aa(option(A),$o,P,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),Mb),F2))
    <=> ( ( ( Mb = none(B) )
         => aa(option(A),$o,P,none(A)) )
        & ! [V5: B] :
            ( ( Mb = aa(B,option(B),some(B),V5) )
           => aa(option(A),$o,P,aa(B,option(A),F2,V5)) ) ) ) ).

% bind_split
tff(fact_8056_map__option__bind,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(B,A),Xb: option(C),G: fun(C,option(B))] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),aa(fun(C,option(B)),option(B),aa(option(C),fun(fun(C,option(B)),option(B)),bind(C,B),Xb),G)) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),Xb),aa(fun(C,option(B)),fun(C,option(A)),comp(option(B),option(A),C,aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2)),G)) ).

% map_option_bind
tff(fact_8057_bind__map__option,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,B),Xb: option(C),G: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F2),Xb)),G) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),Xb),aa(fun(C,B),fun(C,option(A)),comp(B,option(A),C,G),F2)) ).

% bind_map_option
tff(fact_8058_continuous__attains__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F2)
             => ? [X4: A] :
                  ( member(A,X4,S)
                  & ! [Xa: A] :
                      ( member(A,Xa,S)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,X4)) ) ) ) ) ) ) ).

% continuous_attains_sup
tff(fact_8059_set__bind__option,axiom,
    ! [A: $tType,B: $tType,Xb: option(B),F2: fun(B,option(A))] : aa(option(A),set(A),set_option(A),aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),Xb),F2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),aa(fun(B,option(A)),fun(B,set(A)),comp(option(A),set(A),B,set_option(A)),F2),aa(option(B),set(B),set_option(B),Xb))) ).

% set_bind_option
tff(fact_8060_combine__options__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Xb: option(A),Y: option(A)] : combine_options(A,F2,Xb,Y) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),Y),aa(option(A),fun(A,option(A)),aTP_Lamp_ahu(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),F2),Y)),Xb) ).

% combine_options_def
tff(fact_8061_elem__set,axiom,
    ! [A: $tType,Xb: A,Xo: option(A)] :
      ( member(A,Xb,aa(option(A),set(A),set_option(A),Xo))
    <=> ( Xo = aa(A,option(A),some(A),Xb) ) ) ).

% elem_set
tff(fact_8062_combine__options__simps_I3_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A] : combine_options(A,F2,aa(A,option(A),some(A),A3),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F2,A3),B2)) ).

% combine_options_simps(3)
tff(fact_8063_combine__options__simps_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Y: option(A)] : combine_options(A,F2,none(A),Y) = Y ).

% combine_options_simps(1)
tff(fact_8064_combine__options__simps_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Xb: option(A)] : combine_options(A,F2,Xb,none(A)) = Xb ).

% combine_options_simps(2)
tff(fact_8065_set__empty__eq,axiom,
    ! [A: $tType,Xo: option(A)] :
      ( ( aa(option(A),set(A),set_option(A),Xo) = bot_bot(set(A)) )
    <=> ( Xo = none(A) ) ) ).

% set_empty_eq
tff(fact_8066_bind__option__cong,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),Y: option(A),F2: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( Xb = Y )
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(option(A),set(A),set_option(A),Y))
           => ( aa(A,option(B),F2,Z3) = aa(A,option(B),G,Z3) ) )
       => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Xb),F2) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y),G) ) ) ) ).

% bind_option_cong
tff(fact_8067_option_Oset__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),V: option(B)] : aa(option(A),set(A),set_option(A),aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),V)) = image(B,A,F2,aa(option(B),set(B),set_option(B),V)) ).

% option.set_map
tff(fact_8068_option_Oset__sel,axiom,
    ! [A: $tType,A3: option(A)] :
      ( ( A3 != none(A) )
     => member(A,aa(option(A),A,the2(A),A3),aa(option(A),set(A),set_option(A),A3)) ) ).

% option.set_sel
tff(fact_8069_ospec,axiom,
    ! [A: $tType,A4: option(A),P: fun(A,$o),Xb: A] :
      ( ! [X4: A] :
          ( member(A,X4,aa(option(A),set(A),set_option(A),A4))
         => aa(A,$o,P,X4) )
     => ( ( A4 = aa(A,option(A),some(A),Xb) )
       => aa(A,$o,P,Xb) ) ) ).

% ospec
tff(fact_8070_option_Oset__intros,axiom,
    ! [A: $tType,X22: A] : member(A,X22,aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X22))) ).

% option.set_intros
tff(fact_8071_option_Oset__cases,axiom,
    ! [A: $tType,E: A,A3: option(A)] :
      ( member(A,E,aa(option(A),set(A),set_option(A),A3))
     => ( A3 = aa(A,option(A),some(A),E) ) ) ).

% option.set_cases
tff(fact_8072_combine__options__left__commute,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Y: option(A),Xb: option(A),Z: option(A)] :
      ( ! [X4: A,Y4: A] : aa(A,A,aa(A,fun(A,A),F2,X4),Y4) = aa(A,A,aa(A,fun(A,A),F2,Y4),X4)
     => ( ! [X4: A,Y4: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,X4),Y4)),Z3) = aa(A,A,aa(A,fun(A,A),F2,X4),aa(A,A,aa(A,fun(A,A),F2,Y4),Z3))
       => ( combine_options(A,F2,Y,combine_options(A,F2,Xb,Z)) = combine_options(A,F2,Xb,combine_options(A,F2,Y,Z)) ) ) ) ).

% combine_options_left_commute
tff(fact_8073_combine__options__commute,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Xb: option(A),Y: option(A)] :
      ( ! [X4: A,Y4: A] : aa(A,A,aa(A,fun(A,A),F2,X4),Y4) = aa(A,A,aa(A,fun(A,A),F2,Y4),X4)
     => ( combine_options(A,F2,Xb,Y) = combine_options(A,F2,Y,Xb) ) ) ).

% combine_options_commute
tff(fact_8074_combine__options__assoc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Xb: option(A),Y: option(A),Z: option(A)] :
      ( ! [X4: A,Y4: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,X4),Y4)),Z3) = aa(A,A,aa(A,fun(A,A),F2,X4),aa(A,A,aa(A,fun(A,A),F2,Y4),Z3))
     => ( combine_options(A,F2,combine_options(A,F2,Xb,Y),Z) = combine_options(A,F2,Xb,combine_options(A,F2,Y,Z)) ) ) ).

% combine_options_assoc
tff(fact_8075_map__option__idI,axiom,
    ! [A: $tType,Xb: option(A),F2: fun(A,A)] :
      ( ! [Y4: A] :
          ( member(A,Y4,aa(option(A),set(A),set_option(A),Xb))
         => ( aa(A,A,F2,Y4) = Y4 ) )
     => ( aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),F2),Xb) = Xb ) ) ).

% map_option_idI
tff(fact_8076_option_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),Xaa: option(A),F2: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z3: A,Za2: A] :
          ( member(A,Z3,aa(option(A),set(A),set_option(A),Xb))
         => ( member(A,Za2,aa(option(A),set(A),set_option(A),Xaa))
           => ( ( aa(A,B,F2,Z3) = aa(A,B,Fa,Za2) )
             => ( Z3 = Za2 ) ) ) )
     => ( ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),Xb) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),Fa),Xaa) )
       => ( Xb = Xaa ) ) ) ).

% option.inj_map_strong
tff(fact_8077_option_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [Z3: A] :
          ( member(A,Z3,aa(option(A),set(A),set_option(A),Xb))
         => ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
     => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),Xb) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Xb) ) ) ).

% option.map_cong0
tff(fact_8078_option_Omap__cong,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),Ya: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xb = Ya )
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(option(A),set(A),set_option(A),Ya))
           => ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),Xb) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Ya) ) ) ) ).

% option.map_cong
tff(fact_8079_option_Osimps_I14_J,axiom,
    ! [A: $tType] : aa(option(A),set(A),set_option(A),none(A)) = bot_bot(set(A)) ).

% option.simps(14)
tff(fact_8080_option_Osimps_I15_J,axiom,
    ! [A: $tType,X22: A] : aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X22)) = aa(set(A),set(A),insert(A,X22),bot_bot(set(A))) ).

% option.simps(15)
tff(fact_8081_option_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: option(A),B2: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),A3),B2)
    <=> ? [Z2: option(product_prod(A,B))] :
          ( member(option(product_prod(A,B)),Z2,aa(fun(option(product_prod(A,B)),$o),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_ahv(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),R2)))
          & ( aa(option(product_prod(A,B)),option(A),aa(fun(product_prod(A,B),A),fun(option(product_prod(A,B)),option(A)),map_option(product_prod(A,B),A),product_fst(A,B)),Z2) = A3 )
          & ( aa(option(product_prod(A,B)),option(B),aa(fun(product_prod(A,B),B),fun(option(product_prod(A,B)),option(B)),map_option(product_prod(A,B),B),product_snd(A,B)),Z2) = B2 ) ) ) ).

% option.in_rel
tff(fact_8082_cut__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),R2: set(product_prod(A,A)),Xb: A,X5: A] :
      aa(A,B,cut(A,B,F2,R2,Xb),X5) = $ite(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xb),R2),aa(A,B,F2,X5),undefined(B)) ).

% cut_def
tff(fact_8083_rel__option__None2,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Xb: option(A)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),P),Xb),none(B))
    <=> ( Xb = none(A) ) ) ).

% rel_option_None2
tff(fact_8084_rel__option__None1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xb: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),P),none(A)),Xb)
    <=> ( Xb = none(B) ) ) ).

% rel_option_None1
tff(fact_8085_rel__option__reflI,axiom,
    ! [A: $tType,Y: option(A),P: fun(A,fun(A,$o))] :
      ( ! [X4: A] :
          ( member(A,X4,aa(option(A),set(A),set_option(A),Y))
         => aa(A,$o,aa(A,fun(A,$o),P,X4),X4) )
     => aa(option(A),$o,aa(option(A),fun(option(A),$o),aa(fun(A,fun(A,$o)),fun(option(A),fun(option(A),$o)),rel_option(A,A),P),Y),Y) ) ).

% rel_option_reflI
tff(fact_8086_option_Orel__refl__strong,axiom,
    ! [A: $tType,Xb: option(A),Ra2: fun(A,fun(A,$o))] :
      ( ! [Z3: A] :
          ( member(A,Z3,aa(option(A),set(A),set_option(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),Ra2,Z3),Z3) )
     => aa(option(A),$o,aa(option(A),fun(option(A),$o),aa(fun(A,fun(A,$o)),fun(option(A),fun(option(A),$o)),rel_option(A,A),Ra2),Xb),Xb) ) ).

% option.rel_refl_strong
tff(fact_8087_option_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Xb: option(A),Y: option(B),Ra2: fun(A,fun(B,$o))] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),Xb),Y)
     => ( ! [Z3: A,Yb2: B] :
            ( member(A,Z3,aa(option(A),set(A),set_option(A),Xb))
           => ( member(B,Yb2,aa(option(B),set(B),set_option(B),Y))
             => ( aa(B,$o,aa(A,fun(B,$o),R2,Z3),Yb2)
               => aa(B,$o,aa(A,fun(B,$o),Ra2,Z3),Yb2) ) ) )
       => aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),Ra2),Xb),Y) ) ) ).

% option.rel_mono_strong
tff(fact_8088_option_Orel__cong,axiom,
    ! [A: $tType,B: $tType,Xb: option(A),Ya: option(A),Y: option(B),Xaa: option(B),R2: fun(A,fun(B,$o)),Ra2: fun(A,fun(B,$o))] :
      ( ( Xb = Ya )
     => ( ( Y = Xaa )
       => ( ! [Z3: A,Yb2: B] :
              ( member(A,Z3,aa(option(A),set(A),set_option(A),Ya))
             => ( member(B,Yb2,aa(option(B),set(B),set_option(B),Xaa))
               => ( aa(B,$o,aa(A,fun(B,$o),R2,Z3),Yb2)
                <=> aa(B,$o,aa(A,fun(B,$o),Ra2,Z3),Yb2) ) ) )
         => ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),Xb),Y)
          <=> aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),Ra2),Ya),Xaa) ) ) ) ) ).

% option.rel_cong
tff(fact_8089_option_OQuotient,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(A,$o)),Abs: fun(A,B),Rep: fun(B,A),T5: fun(A,fun(B,$o))] :
      ( quotient(A,B,R2,Abs,Rep,T5)
     => quotient(option(A),option(B),aa(fun(A,fun(A,$o)),fun(option(A),fun(option(A),$o)),rel_option(A,A),R2),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),Abs),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),Rep),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),T5)) ) ).

% option.Quotient
tff(fact_8090_option_Orel__conversep,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,fun(A,$o))] : aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),conversep(B,A,R2)) = conversep(option(B),option(A),aa(fun(B,fun(A,$o)),fun(option(B),fun(option(A),$o)),rel_option(B,A),R2)) ).

% option.rel_conversep
tff(fact_8091_option_Orel__flip,axiom,
    ! [B: $tType,A: $tType,R2: fun(B,fun(A,$o)),A3: option(A),B2: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),conversep(B,A,R2)),A3),B2)
    <=> aa(option(A),$o,aa(option(B),fun(option(A),$o),aa(fun(B,fun(A,$o)),fun(option(B),fun(option(A),$o)),rel_option(B,A),R2),B2),A3) ) ).

% option.rel_flip
tff(fact_8092_option_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o)),Ra2: 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,$o))),R2),Ra2)
     => aa(fun(option(A),fun(option(B),$o)),$o,aa(fun(option(A),fun(option(B),$o)),fun(fun(option(A),fun(option(B),$o)),$o),ord_less_eq(fun(option(A),fun(option(B),$o))),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2)),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),Ra2)) ) ).

% option.rel_mono
tff(fact_8093_option_Orel__transp,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( transp(A,R2)
     => transp(option(A),aa(fun(A,fun(A,$o)),fun(option(A),fun(option(A),$o)),rel_option(A,A),R2)) ) ).

% option.rel_transp
tff(fact_8094_option_Orec__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,S3: fun(A,fun(B,$o)),R2: fun(C,fun(D,$o))] : aa(fun(B,fun(fun(D,B),fun(option(D),B))),$o,aa(fun(A,fun(fun(C,A),fun(option(C),A))),fun(fun(B,fun(fun(D,B),fun(option(D),B))),$o),bNF_rel_fun(A,B,fun(fun(C,A),fun(option(C),A)),fun(fun(D,B),fun(option(D),B)),S3,bNF_rel_fun(fun(C,A),fun(D,B),fun(option(C),A),fun(option(D),B),bNF_rel_fun(C,D,A,B,R2,S3),bNF_rel_fun(option(C),option(D),A,B,aa(fun(C,fun(D,$o)),fun(option(C),fun(option(D),$o)),rel_option(C,D),R2),S3))),rec_option(A,C)),rec_option(B,D)) ).

% option.rec_transfer
tff(fact_8095_option_Omap__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Rb2: fun(A,fun(C,$o)),Sd: fun(B,fun(D,$o))] : aa(fun(fun(C,D),fun(option(C),option(D))),$o,aa(fun(fun(A,B),fun(option(A),option(B))),fun(fun(fun(C,D),fun(option(C),option(D))),$o),bNF_rel_fun(fun(A,B),fun(C,D),fun(option(A),option(B)),fun(option(C),option(D)),bNF_rel_fun(A,C,B,D,Rb2,Sd),bNF_rel_fun(option(A),option(C),option(B),option(D),aa(fun(A,fun(C,$o)),fun(option(A),fun(option(C),$o)),rel_option(A,C),Rb2),aa(fun(B,fun(D,$o)),fun(option(B),fun(option(D),$o)),rel_option(B,D),Sd))),map_option(A,B)),map_option(C,D)) ).

% option.map_transfer
tff(fact_8096_option_Orel__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,$o)),Sc: fun(B,fun(D,$o))] : aa(fun(fun(C,fun(D,$o)),fun(option(C),fun(option(D),$o))),$o,aa(fun(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o))),fun(fun(fun(C,fun(D,$o)),fun(option(C),fun(option(D),$o))),$o),bNF_rel_fun(fun(A,fun(B,$o)),fun(C,fun(D,$o)),fun(option(A),fun(option(B),$o)),fun(option(C),fun(option(D),$o)),bNF_rel_fun(A,C,fun(B,$o),fun(D,$o),Sa,bNF_rel_fun(B,D,$o,$o,Sc,fequal($o))),bNF_rel_fun(option(A),option(C),fun(option(B),$o),fun(option(D),$o),aa(fun(A,fun(C,$o)),fun(option(A),fun(option(C),$o)),rel_option(A,C),Sa),bNF_rel_fun(option(B),option(D),$o,$o,aa(fun(B,fun(D,$o)),fun(option(B),fun(option(D),$o)),rel_option(B,D),Sc),fequal($o)))),rel_option(A,B)),rel_option(C,D)) ).

% option.rel_transfer
tff(fact_8097_option_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] : aa(fun(option(B),$o),$o,aa(fun(option(A),$o),fun(fun(option(B),$o),$o),bNF_rel_fun(option(A),option(B),$o,$o,aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),fequal($o)),aTP_Lamp_ahw(option(A),$o)),aTP_Lamp_ahx(option(B),$o)) ).

% option.disc_transfer(1)
tff(fact_8098_option_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] : aa(fun(option(B),$o),$o,aa(fun(option(A),$o),fun(fun(option(B),$o),$o),bNF_rel_fun(option(A),option(B),$o,$o,aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),fequal($o)),aTP_Lamp_ahy(option(A),$o)),aTP_Lamp_ahz(option(B),$o)) ).

% option.disc_transfer(2)
tff(fact_8099_option_Octr__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] : aa(fun(B,option(B)),$o,aa(fun(A,option(A)),fun(fun(B,option(B)),$o),bNF_rel_fun(A,B,option(A),option(B),R2,aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2)),some(A)),some(B)) ).

% option.ctr_transfer(2)
tff(fact_8100_option_Ocase__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,S3: fun(A,fun(B,$o)),R2: fun(C,fun(D,$o))] : aa(fun(B,fun(fun(D,B),fun(option(D),B))),$o,aa(fun(A,fun(fun(C,A),fun(option(C),A))),fun(fun(B,fun(fun(D,B),fun(option(D),B))),$o),bNF_rel_fun(A,B,fun(fun(C,A),fun(option(C),A)),fun(fun(D,B),fun(option(D),B)),S3,bNF_rel_fun(fun(C,A),fun(D,B),fun(option(C),A),fun(option(D),B),bNF_rel_fun(C,D,A,B,R2,S3),bNF_rel_fun(option(C),option(D),A,B,aa(fun(C,fun(D,$o)),fun(option(C),fun(option(D),$o)),rel_option(C,D),R2),S3))),case_option(A,C)),case_option(B,D)) ).

% option.case_transfer
tff(fact_8101_rel__option__inf,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,fun(B,$o)),B4: fun(A,fun(B,$o))] : aa(fun(option(A),fun(option(B),$o)),fun(option(A),fun(option(B),$o)),aa(fun(option(A),fun(option(B),$o)),fun(fun(option(A),fun(option(B),$o)),fun(option(A),fun(option(B),$o))),inf_inf(fun(option(A),fun(option(B),$o))),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),A4)),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),B4)) = aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),aa(fun(A,fun(B,$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))),A4),B4)) ).

% rel_option_inf
tff(fact_8102_option_Obi__total__rel,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] :
      ( bi_total(A,B,R2)
     => bi_total(option(A),option(B),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2)) ) ).

% option.bi_total_rel
tff(fact_8103_option_Orel__sel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: option(A),B2: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),A3),B2)
    <=> ( ( ( A3 = none(A) )
        <=> ( B2 = none(B) ) )
        & ( ( A3 != none(A) )
         => ( ( B2 != none(B) )
           => aa(B,$o,aa(A,fun(B,$o),R2,aa(option(A),A,the2(A),A3)),aa(option(B),B,the2(B),B2)) ) ) ) ) ).

% option.rel_sel
tff(fact_8104_option_Orel__induct,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Xb: option(A),Y: option(B),Q: fun(option(A),fun(option(B),$o))] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),Xb),Y)
     => ( aa(option(B),$o,aa(option(A),fun(option(B),$o),Q,none(A)),none(B))
       => ( ! [A23: A,B23: B] :
              ( aa(B,$o,aa(A,fun(B,$o),R2,A23),B23)
             => aa(option(B),$o,aa(option(A),fun(option(B),$o),Q,aa(A,option(A),some(A),A23)),aa(B,option(B),some(B),B23)) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),Q,Xb),Y) ) ) ) ).

% option.rel_induct
tff(fact_8105_option_Orel__cases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: option(A),B2: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),A3),B2)
     => ( ( ( A3 = none(A) )
         => ( B2 != none(B) ) )
       => ~ ! [X4: A] :
              ( ( A3 = aa(A,option(A),some(A),X4) )
             => ! [Y4: B] :
                  ( ( B2 = aa(B,option(B),some(B),Y4) )
                 => ~ aa(B,$o,aa(A,fun(B,$o),R2,X4),Y4) ) ) ) ) ).

% option.rel_cases
tff(fact_8106_option_Orel__distinct_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Y2: B] : ~ aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),none(A)),aa(B,option(B),some(B),Y2)) ).

% option.rel_distinct(1)
tff(fact_8107_option_Orel__distinct_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Y2: A] : ~ aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),aa(A,option(A),some(A),Y2)),none(B)) ).

% option.rel_distinct(2)
tff(fact_8108_option_Orel__map_I2_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,Sa: fun(A,fun(B,$o)),Xb: option(A),G: fun(C,B),Y: option(C)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),Sa),Xb),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),G),Y))
    <=> aa(option(C),$o,aa(option(A),fun(option(C),$o),aa(fun(A,fun(C,$o)),fun(option(A),fun(option(C),$o)),rel_option(A,C),aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aia(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),Sa),G)),Xb),Y) ) ).

% option.rel_map(2)
tff(fact_8109_option_Orel__map_I1_J,axiom,
    ! [C: $tType,A: $tType,B: $tType,Sb: fun(A,fun(B,$o)),I: fun(C,A),Xb: option(C),Y: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),Sb),aa(option(C),option(A),aa(fun(C,A),fun(option(C),option(A)),map_option(C,A),I),Xb)),Y)
    <=> aa(option(B),$o,aa(option(C),fun(option(B),$o),aa(fun(C,fun(B,$o)),fun(option(C),fun(option(B),$o)),rel_option(C,B),aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_aib(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),Sb),I)),Xb),Y) ) ).

% option.rel_map(1)
tff(fact_8110_option_Orel__eq,axiom,
    ! [A: $tType] : aa(fun(A,fun(A,$o)),fun(option(A),fun(option(A),$o)),rel_option(A,A),fequal(A)) = fequal(option(A)) ).

% option.rel_eq
tff(fact_8111_option_Orel__refl,axiom,
    ! [A: $tType,Ra2: fun(A,fun(A,$o)),Xb: option(A)] :
      ( ! [X4: A] : aa(A,$o,aa(A,fun(A,$o),Ra2,X4),X4)
     => aa(option(A),$o,aa(option(A),fun(option(A),$o),aa(fun(A,fun(A,$o)),fun(option(A),fun(option(A),$o)),rel_option(A,A),Ra2),Xb),Xb) ) ).

% option.rel_refl
tff(fact_8112_option_Orel__inject_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),X22: A,Y2: B] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),aa(A,option(A),some(A),X22)),aa(B,option(B),some(B),Y2))
    <=> aa(B,$o,aa(A,fun(B,$o),R2,X22),Y2) ) ).

% option.rel_inject(2)
tff(fact_8113_option_Orel__intros_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),X22: A,Y2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,X22),Y2)
     => aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),aa(A,option(A),some(A),X22)),aa(B,option(B),some(B),Y2)) ) ).

% option.rel_intros(2)
tff(fact_8114_option__rel__Some1,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),Xb: A,Y: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),A4),aa(A,option(A),some(A),Xb)),Y)
    <=> ? [Y8: B] :
          ( ( Y = aa(B,option(B),some(B),Y8) )
          & aa(B,$o,aa(A,fun(B,$o),A4,Xb),Y8) ) ) ).

% option_rel_Some1
tff(fact_8115_option__rel__Some2,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),Xb: option(A),Y: B] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),A4),Xb),aa(B,option(B),some(B),Y))
    <=> ? [X10: A] :
          ( ( Xb = aa(A,option(A),some(A),X10) )
          & aa(B,$o,aa(A,fun(B,$o),A4,X10),Y) ) ) ).

% option_rel_Some2
tff(fact_8116_option_Octr__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] : aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),none(A)),none(B)) ).

% option.ctr_transfer(1)
tff(fact_8117_rel__option__iff,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Xb: option(A),Y: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),Xb),Y)
    <=> aa(product_prod(option(A),option(B)),$o,aa(fun(option(A),fun(option(B),$o)),fun(product_prod(option(A),option(B)),$o),product_case_prod(option(A),option(B),$o),aTP_Lamp_aie(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),R2)),aa(option(B),product_prod(option(A),option(B)),aa(option(A),fun(option(B),product_prod(option(A),option(B))),product_Pair(option(A),option(B)),Xb),Y)) ) ).

% rel_option_iff
tff(fact_8118_cuts__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),R2: set(product_prod(A,A)),Xb: A,G: fun(A,B)] :
      ( ( cut(A,B,F2,R2,Xb) = cut(A,B,G,R2,Xb) )
    <=> ! [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),Xb),R2)
         => ( aa(A,B,F2,Y3) = aa(A,B,G,Y3) ) ) ) ).

% cuts_eq
tff(fact_8119_cut__apply,axiom,
    ! [B: $tType,A: $tType,Xb: A,A3: A,R2: set(product_prod(A,A)),F2: fun(A,B)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),A3),R2)
     => ( aa(A,B,cut(A,B,F2,R2,A3),Xb) = aa(A,B,F2,Xb) ) ) ).

% cut_apply
tff(fact_8120_option__bind__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,$o)),B4: fun(C,fun(D,$o))] : aa(fun(option(B),fun(fun(B,option(D)),option(D))),$o,aa(fun(option(A),fun(fun(A,option(C)),option(C))),fun(fun(option(B),fun(fun(B,option(D)),option(D))),$o),bNF_rel_fun(option(A),option(B),fun(fun(A,option(C)),option(C)),fun(fun(B,option(D)),option(D)),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),A4),bNF_rel_fun(fun(A,option(C)),fun(B,option(D)),option(C),option(D),bNF_rel_fun(A,B,option(C),option(D),A4,aa(fun(C,fun(D,$o)),fun(option(C),fun(option(D),$o)),rel_option(C,D),B4)),aa(fun(C,fun(D,$o)),fun(option(C),fun(option(D),$o)),rel_option(C,D),B4))),bind(A,C)),bind(B,D)) ).

% option_bind_transfer
tff(fact_8121_option_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] : aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2) = relcompp(option(A),option(product_prod(A,B)),option(B),conversep(option(product_prod(A,B)),option(A),bNF_Grp(option(product_prod(A,B)),option(A),aa(fun(option(product_prod(A,B)),$o),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_ahv(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),R2)),aa(fun(product_prod(A,B),A),fun(option(product_prod(A,B)),option(A)),map_option(product_prod(A,B),A),product_fst(A,B)))),bNF_Grp(option(product_prod(A,B)),option(B),aa(fun(option(product_prod(A,B)),$o),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_ahv(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),R2)),aa(fun(product_prod(A,B),B),fun(option(product_prod(A,B)),option(B)),map_option(product_prod(A,B),B),product_snd(A,B)))) ).

% option.rel_compp_Grp
tff(fact_8122_above__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_above(A,R,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,$o)),R),A3)) ).

% above_def
tff(fact_8123_option_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B)] : aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),bNF_Grp(A,B,A4,F2)) = bNF_Grp(option(A),option(B),aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_aif(set(A),fun(option(A),$o),A4)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2)) ).

% option.rel_Grp
tff(fact_8124_option_Orel__compp,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(A,fun(C,$o)),S3: fun(C,fun(B,$o))] : aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),relcompp(A,C,B,R2,S3)) = relcompp(option(A),option(C),option(B),aa(fun(A,fun(C,$o)),fun(option(A),fun(option(C),$o)),rel_option(A,C),R2),aa(fun(C,fun(B,$o)),fun(option(C),fun(option(B),$o)),rel_option(C,B),S3)) ).

% option.rel_compp
tff(fact_8125_relcompp__relcomp__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: set(product_prod(A,C)),S: set(product_prod(C,B)),X5: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),relcompp(A,C,B,aTP_Lamp_aig(set(product_prod(A,C)),fun(A,fun(C,$o)),R),aTP_Lamp_aih(set(product_prod(C,B)),fun(C,fun(B,$o)),S)),X5),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa),relcomp(A,C,B,R,S)) ) ).

% relcompp_relcomp_eq
tff(fact_8126_relcomp__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,X5: set(product_prod(A,B)),Xa: set(product_prod(B,C))] : relcomp(A,B,C,X5,Xa) = aa(fun(product_prod(A,C),$o),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,$o)),fun(product_prod(A,C),$o),product_case_prod(A,C,$o),relcompp(A,B,C,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),X5),aTP_Lamp_aii(set(product_prod(B,C)),fun(B,fun(C,$o)),Xa)))) ).

% relcomp_def
tff(fact_8127_relpowp__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,P: fun(A,fun(A,$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,$o))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),P) = relcompp(A,A,A,aa(fun(A,fun(A,$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))),Mb),P),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)) ).

% relpowp_add
tff(fact_8128_option_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(option(B),$o)),$o,aa(fun(fun(A,$o),fun(option(A),$o)),fun(fun(fun(B,$o),fun(option(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(option(A),$o),fun(option(B),$o),bNF_rel_fun(A,B,$o,$o,R2,fequal($o)),bNF_rel_fun(option(A),option(B),$o,$o,aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2),fequal($o))),pred_option(A)),pred_option(B)) ).

% option.pred_transfer
tff(fact_8129_pred__option__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(option(B),$o)),$o,aa(fun(fun(A,$o),fun(option(A),$o)),fun(fun(fun(B,$o),fun(option(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(option(A),$o),fun(option(B),$o),bNF_rel_fun(A,B,$o,$o,A4,fequal($o)),bNF_rel_fun(option(A),option(B),$o,$o,aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),A4),fequal($o))),pred_option(A)),pred_option(B)) ).

% pred_option_parametric
tff(fact_8130_option_Opred__inject_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),P),aa(A,option(A),some(A),A3))
    <=> aa(A,$o,P,A3) ) ).

% option.pred_inject(2)
tff(fact_8131_pcr__real__def,axiom,
    pcr_real = relcompp(fun(nat,rat),fun(nat,rat),real,bNF_rel_fun(nat,nat,rat,rat,fequal(nat),fequal(rat)),cr_real) ).

% pcr_real_def
tff(fact_8132_option_Opred__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Pa: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Pa)
     => aa(fun(option(A),$o),$o,aa(fun(option(A),$o),fun(fun(option(A),$o),$o),ord_less_eq(fun(option(A),$o)),aa(fun(A,$o),fun(option(A),$o),pred_option(A),P)),aa(fun(A,$o),fun(option(A),$o),pred_option(A),Pa)) ) ).

% option.pred_mono
tff(fact_8133_option_Opred__True,axiom,
    ! [A: $tType,X5: option(A)] : aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),aTP_Lamp_lv(A,$o)),X5) ).

% option.pred_True
tff(fact_8134_option_Opred__inject_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),P),none(A)) ).

% option.pred_inject(1)
tff(fact_8135_option_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),Ya: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xb = Ya )
     => ( aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_aij(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G)),Ya)
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),Xb) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Ya) ) ) ) ).

% option.map_cong_pred
tff(fact_8136_option_Opred__cong,axiom,
    ! [A: $tType,Xb: option(A),Ya: option(A),P: fun(A,$o),Pa: fun(A,$o)] :
      ( ( Xb = Ya )
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(option(A),set(A),set_option(A),Ya))
           => ( aa(A,$o,P,Z3)
            <=> aa(A,$o,Pa,Z3) ) )
       => ( aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),P),Xb)
        <=> aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),Pa),Ya) ) ) ) ).

% option.pred_cong
tff(fact_8137_option_Opred__mono__strong,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: option(A),Pa: fun(A,$o)] :
      ( aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),P),Xb)
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(option(A),set(A),set_option(A),Xb))
           => ( aa(A,$o,P,Z3)
             => aa(A,$o,Pa,Z3) ) )
       => aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),Pa),Xb) ) ) ).

% option.pred_mono_strong
tff(fact_8138_option_Opred__set,axiom,
    ! [A: $tType,P: fun(A,$o),X5: option(A)] :
      ( aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),P),X5)
    <=> ! [Xa2: A] :
          ( member(A,Xa2,aa(option(A),set(A),set_option(A),X5))
         => aa(A,$o,P,Xa2) ) ) ).

% option.pred_set
tff(fact_8139_option_Opred__map,axiom,
    ! [A: $tType,B: $tType,Q: fun(A,$o),F2: fun(B,A),Xb: option(B)] :
      ( aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),pred_option(A),Q),aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),Xb))
    <=> aa(option(B),$o,aa(fun(B,$o),fun(option(B),$o),pred_option(B),aa(fun(B,A),fun(B,$o),comp(A,$o,B,Q),F2)),Xb) ) ).

% option.pred_map
tff(fact_8140_internalize__ordLess,axiom,
    ! [A: $tType,B: $tType,R4: set(product_prod(A,A)),R: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R4),R),bNF_We4044943003108391690rdLess(A,B))
    <=> ? [P7: set(product_prod(B,B))] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),field2(B,P7)),field2(B,R))
          & member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R4),P7),bNF_Wellorder_ordIso(A,B))
          & member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P7),R),bNF_We4044943003108391690rdLess(B,B)) ) ) ).

% internalize_ordLess
tff(fact_8141_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)),removeAll(A,Xb,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_removeAll_less
tff(fact_8142_length__removeAll__less__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,Xb,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_removeAll_less_eq
tff(fact_8143_real_Odomain,axiom,
    ! [X5: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,domainp(fun(nat,rat),real,pcr_real),X5)
    <=> ? [Y3: fun(nat,rat)] :
          ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),bNF_rel_fun(nat,nat,rat,rat,fequal(nat),fequal(rat)),X5),Y3)
          & aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Y3),Y3) ) ) ).

% real.domain
tff(fact_8144_lcm__altdef__int,axiom,
    ! [A3: int,B2: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A3),B2) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A3)),aa(int,int,abs_abs(int),B2))),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)) ).

% lcm_altdef_int
tff(fact_8145_lcm__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: num,A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),A3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(num,A,numeral_numeral(A),Nb)),A3) ) ).

% lcm_neg_numeral_1
tff(fact_8146_lcm__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,Nb: num] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(num,A,numeral_numeral(A),Nb)) ) ).

% lcm_neg_numeral_2
tff(fact_8147_lcm__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = one_one(A) )
        <=> ( dvd_dvd(A,A3,one_one(A))
            & dvd_dvd(A,B2,one_one(A)) ) ) ) ).

% lcm_eq_1_iff
tff(fact_8148_Domainp__pcr__real,axiom,
    domainp(fun(nat,rat),real,pcr_real) = cauchy ).

% Domainp_pcr_real
tff(fact_8149_real_Odomain__eq,axiom,
    ! [X5: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,domainp(fun(nat,rat),real,pcr_real),X5)
    <=> aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X5),X5) ) ).

% real.domain_eq
tff(fact_8150_lcm__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_mult_unit1
tff(fact_8151_lcm__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_mult_unit2
tff(fact_8152_lcm__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_div_unit2
tff(fact_8153_lcm__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_div_unit1
tff(fact_8154_lcm__pos__int,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero(int) )
     => ( ( Nb != zero_zero(int) )
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Mb),Nb)) ) ) ).

% lcm_pos_int
tff(fact_8155_Domainp__Domain__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),X5: A] :
      ( aa(A,$o,domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),X5)
    <=> member(A,X5,domain(A,B,R)) ) ).

% Domainp_Domain_eq
tff(fact_8156_Domain__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : domain(A,B,X5) = aa(fun(A,$o),set(A),collect(A),domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),X5))) ).

% Domain_def
tff(fact_8157_option_ODomainp__rel,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] : domainp(option(A),option(B),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2)) = aa(fun(A,$o),fun(option(A),$o),pred_option(A),domainp(A,B,R2)) ).

% option.Domainp_rel
tff(fact_8158_real_Odomain__par__left__total,axiom,
    ! [P3: fun(fun(nat,rat),$o)] :
      ( left_total(fun(nat,rat),fun(nat,rat),bNF_rel_fun(nat,nat,rat,rat,fequal(nat),fequal(rat)))
     => ( aa(fun(fun(nat,rat),$o),$o,aa(fun(fun(nat,rat),$o),fun(fun(fun(nat,rat),$o),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),$o,$o,bNF_rel_fun(nat,nat,rat,rat,fequal(nat),fequal(rat)),fequal($o)),P3),aTP_Lamp_aik(fun(nat,rat),$o))
       => ( domainp(fun(nat,rat),real,pcr_real) = P3 ) ) ) ).

% real.domain_par_left_total
tff(fact_8159_Lcm__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = $ite(finite_finite(A,A4),finite_fold(A,A,gcd_lcm(A),one_one(A),A4),zero_zero(A)) ) ).

% Lcm_fin.eq_fold
tff(fact_8160_Lcm__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_fin.empty
tff(fact_8161_is__unit__Lcm__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( dvd_dvd(A,aa(set(A),A,semiring_gcd_Lcm_fin(A),A4),one_one(A))
        <=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = one_one(A) ) ) ) ).

% is_unit_Lcm_fin_iff
tff(fact_8162_lcm__nat__def,axiom,
    ! [Xb: nat,Y: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Xb),Y) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xb),Y)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Y)) ).

% lcm_nat_def
tff(fact_8163_prod__gcd__lcm__nat,axiom,
    ! [Mb: nat,Nb: 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),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Mb),Nb)) ).

% prod_gcd_lcm_nat
tff(fact_8164_lcm__pos__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),zero_zero(nat)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Mb),Nb)) ) ) ).

% lcm_pos_nat
tff(fact_8165_option_Oleft__total__rel,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] :
      ( left_total(A,B,R2)
     => left_total(option(A),option(B),aa(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),rel_option(A,B),R2)) ) ).

% option.left_total_rel
tff(fact_8166_Lcm__fin__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = one_one(A) )
        <=> ( ! [X: A] :
                ( member(A,X,A4)
               => dvd_dvd(A,X,one_one(A)) )
            & finite_finite(A,A4) ) ) ) ).

% Lcm_fin_1_iff
tff(fact_8167_Lcm__fin__def,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Lcm_fin(A) = bounde2362111253966948842tice_F(A,gcd_lcm(A),one_one(A),zero_zero(A)) ) ) ).

% Lcm_fin_def
tff(fact_8168_Rep__real__induct,axiom,
    ! [Y: set(fun(nat,rat)),P: fun(set(fun(nat,rat)),$o)] :
      ( member(set(fun(nat,rat)),Y,aa(fun(set(fun(nat,rat)),$o),set(set(fun(nat,rat))),collect(set(fun(nat,rat))),aTP_Lamp_ail(set(fun(nat,rat)),$o)))
     => ( ! [X4: real] : aa(set(fun(nat,rat)),$o,P,aa(real,set(fun(nat,rat)),rep_real,X4))
       => aa(set(fun(nat,rat)),$o,P,Y) ) ) ).

% Rep_real_induct
tff(fact_8169_Rep__real__cases,axiom,
    ! [Y: set(fun(nat,rat))] :
      ( member(set(fun(nat,rat)),Y,aa(fun(set(fun(nat,rat)),$o),set(set(fun(nat,rat))),collect(set(fun(nat,rat))),aTP_Lamp_ail(set(fun(nat,rat)),$o)))
     => ~ ! [X4: real] : Y != aa(real,set(fun(nat,rat)),rep_real,X4) ) ).

% Rep_real_cases
tff(fact_8170_Rep__real__inject,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,set(fun(nat,rat)),rep_real,Xb) = aa(real,set(fun(nat,rat)),rep_real,Y) )
    <=> ( Xb = Y ) ) ).

% Rep_real_inject
tff(fact_8171_Rep__real,axiom,
    ! [Xb: real] : member(set(fun(nat,rat)),aa(real,set(fun(nat,rat)),rep_real,Xb),aa(fun(set(fun(nat,rat)),$o),set(set(fun(nat,rat))),collect(set(fun(nat,rat))),aTP_Lamp_ail(set(fun(nat,rat)),$o))) ).

% Rep_real
tff(fact_8172_Abs__real__inverse,axiom,
    ! [Y: set(fun(nat,rat))] :
      ( member(set(fun(nat,rat)),Y,aa(fun(set(fun(nat,rat)),$o),set(set(fun(nat,rat))),collect(set(fun(nat,rat))),aTP_Lamp_ail(set(fun(nat,rat)),$o)))
     => ( aa(real,set(fun(nat,rat)),rep_real,aa(set(fun(nat,rat)),real,abs_real,Y)) = Y ) ) ).

% Abs_real_inverse
tff(fact_8173_rep__real__def,axiom,
    rep_real2 = quot_rep(real,fun(nat,rat),rep_real) ).

% rep_real_def
tff(fact_8174_Rep__real__inverse,axiom,
    ! [Xb: real] : aa(set(fun(nat,rat)),real,abs_real,aa(real,set(fun(nat,rat)),rep_real,Xb)) = Xb ).

% Rep_real_inverse
tff(fact_8175_Abs__real__inject,axiom,
    ! [Xb: set(fun(nat,rat)),Y: set(fun(nat,rat))] :
      ( member(set(fun(nat,rat)),Xb,aa(fun(set(fun(nat,rat)),$o),set(set(fun(nat,rat))),collect(set(fun(nat,rat))),aTP_Lamp_ail(set(fun(nat,rat)),$o)))
     => ( member(set(fun(nat,rat)),Y,aa(fun(set(fun(nat,rat)),$o),set(set(fun(nat,rat))),collect(set(fun(nat,rat))),aTP_Lamp_ail(set(fun(nat,rat)),$o)))
       => ( ( aa(set(fun(nat,rat)),real,abs_real,Xb) = aa(set(fun(nat,rat)),real,abs_real,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% Abs_real_inject
tff(fact_8176_Abs__real__induct,axiom,
    ! [P: fun(real,$o),Xb: real] :
      ( ! [Y4: set(fun(nat,rat))] :
          ( member(set(fun(nat,rat)),Y4,aa(fun(set(fun(nat,rat)),$o),set(set(fun(nat,rat))),collect(set(fun(nat,rat))),aTP_Lamp_ail(set(fun(nat,rat)),$o)))
         => aa(real,$o,P,aa(set(fun(nat,rat)),real,abs_real,Y4)) )
     => aa(real,$o,P,Xb) ) ).

% Abs_real_induct
tff(fact_8177_Abs__real__cases,axiom,
    ! [Xb: real] :
      ~ ! [Y4: set(fun(nat,rat))] :
          ( ( Xb = aa(set(fun(nat,rat)),real,abs_real,Y4) )
         => ~ member(set(fun(nat,rat)),Y4,aa(fun(set(fun(nat,rat)),$o),set(set(fun(nat,rat))),collect(set(fun(nat,rat))),aTP_Lamp_ail(set(fun(nat,rat)),$o))) ) ).

% Abs_real_cases
tff(fact_8178_Real__def,axiom,
    real2 = quot_abs(fun(nat,rat),real,realrel,abs_real) ).

% Real_def
tff(fact_8179_type__definition__real,axiom,
    type_definition(real,set(fun(nat,rat)),rep_real,abs_real,aa(fun(set(fun(nat,rat)),$o),set(set(fun(nat,rat))),collect(set(fun(nat,rat))),aTP_Lamp_ail(set(fun(nat,rat)),$o))) ).

% type_definition_real
tff(fact_8180_gcd__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ) ) ) ).

% gcd_lcm
tff(fact_8181_real_Odomain__par,axiom,
    ! [DT: fun(nat,$o),DS: fun(rat,$o),P22: fun(fun(nat,rat),$o)] :
      ( ( domainp(nat,nat,fequal(nat)) = DT )
     => ( ( domainp(rat,rat,fequal(rat)) = DS )
       => ( left_unique(nat,nat,fequal(nat))
         => ( aa(fun(fun(nat,rat),$o),$o,aa(fun(fun(nat,rat),$o),fun(fun(fun(nat,rat),$o),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),$o,$o,bNF_rel_fun(nat,nat,rat,rat,fequal(nat),fequal(rat)),fequal($o)),P22),aTP_Lamp_aik(fun(nat,rat),$o))
           => ( domainp(fun(nat,rat),real,pcr_real) = aa(fun(fun(nat,rat),$o),fun(fun(nat,rat),$o),aa(fun(fun(nat,rat),$o),fun(fun(fun(nat,rat),$o),fun(fun(nat,rat),$o)),inf_inf(fun(fun(nat,rat),$o)),basic_pred_fun(nat,rat,DT,DS)),P22) ) ) ) ) ) ).

% real.domain_par
tff(fact_8182_normalize__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : normal6383669964737779283malize(A,normal6383669964737779283malize(A,A3)) = normal6383669964737779283malize(A,A3) ) ).

% normalize_idem
tff(fact_8183_ATP_Olambda__1,axiom,
    ! [Uu: fun(nat,rat)] :
      aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_xy(fun(nat,rat),fun(nat,rat)),Uu) = $ite(vanishes(Uu),aTP_Lamp_pa(nat,rat),aTP_Lamp_ox(fun(nat,rat),fun(nat,rat),Uu)) ).

% ATP.lambda_1
tff(fact_8184_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_dc(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uu)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat)))) ).

% ATP.lambda_2
tff(fact_8185_ATP_Olambda__3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ug(A,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Uu)),one_one(A))),Uu) ) ).

% ATP.lambda_3
tff(fact_8186_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_ph(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).

% ATP.lambda_4
tff(fact_8187_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_pf(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_5
tff(fact_8188_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_agh(A,$o),Uu)
        <=> ( member(A,Uu,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Uu) ) ) ) ).

% ATP.lambda_6
tff(fact_8189_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_ek(nat,real),Uu) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_7
tff(fact_8190_ATP_Olambda__8,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_im(real,$o),Uu)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_8
tff(fact_8191_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),A,aTP_Lamp_aci(product_prod(int,int),A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_9
tff(fact_8192_ATP_Olambda__10,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ux(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_10
tff(fact_8193_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_db(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu)) ).

% ATP.lambda_11
tff(fact_8194_ATP_Olambda__12,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_uw(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))) ) ).

% ATP.lambda_12
tff(fact_8195_ATP_Olambda__13,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_uj(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,Uu)),sin(real,Uu)) ).

% ATP.lambda_13
tff(fact_8196_ATP_Olambda__14,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: product_prod(A,A)] : aa(product_prod(A,A),A,aTP_Lamp_adi(product_prod(A,A),A),Uu) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(product_prod(A,A),A,product_fst(A,A),Uu)),aa(product_prod(A,A),A,product_snd(A,A),Uu)) ) ).

% ATP.lambda_14
tff(fact_8197_ATP_Olambda__15,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [Uu: product_prod(A,A)] : aa(product_prod(A,A),A,aTP_Lamp_adj(product_prod(A,A),A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(product_prod(A,A),A,product_fst(A,A),Uu)),aa(product_prod(A,A),A,product_snd(A,A),Uu)) ) ).

% ATP.lambda_15
tff(fact_8198_ATP_Olambda__16,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_ags(list(A),product_prod(nat,list(A))),Uu) = aa(list(A),product_prod(nat,list(A)),aa(nat,fun(list(A),product_prod(nat,list(A))),product_Pair(nat,list(A)),aa(list(A),nat,size_size(list(A)),Uu)),Uu) ).

% ATP.lambda_16
tff(fact_8199_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_uq(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_17
tff(fact_8200_ATP_Olambda__18,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_aat(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).

% ATP.lambda_18
tff(fact_8201_ATP_Olambda__19,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_nh(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).

% ATP.lambda_19
tff(fact_8202_ATP_Olambda__20,axiom,
    ! [Uu: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aTP_Lamp_aik(fun(nat,rat),$o),Uu)
    <=> aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Uu),Uu) ) ).

% ATP.lambda_20
tff(fact_8203_ATP_Olambda__21,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_fo(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_21
tff(fact_8204_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_ach(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uu),zero_zero(nat)) ).

% ATP.lambda_22
tff(fact_8205_ATP_Olambda__23,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( aa(option(B),$o,aTP_Lamp_ahx(option(B),$o),Uu)
    <=> ( Uu = none(B) ) ) ).

% ATP.lambda_23
tff(fact_8206_ATP_Olambda__24,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_ahw(option(A),$o),Uu)
    <=> ( Uu = none(A) ) ) ).

% ATP.lambda_24
tff(fact_8207_ATP_Olambda__25,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_acg(product_prod(int,int),$o),Uu)
    <=> 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),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_25
tff(fact_8208_ATP_Olambda__26,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_uv(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_26
tff(fact_8209_ATP_Olambda__27,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ahr(nat,nat),Uu) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu)) ).

% ATP.lambda_27
tff(fact_8210_ATP_Olambda__28,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_oj(list(B),fun(nat,nat)),Uu) = 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)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_28
tff(fact_8211_ATP_Olambda__29,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_abc(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).

% ATP.lambda_29
tff(fact_8212_ATP_Olambda__30,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( aa(option(B),$o,aTP_Lamp_ahz(option(B),$o),Uu)
    <=> ( Uu != none(B) ) ) ).

% ATP.lambda_30
tff(fact_8213_ATP_Olambda__31,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_ahy(option(A),$o),Uu)
    <=> ( Uu != none(A) ) ) ).

% ATP.lambda_31
tff(fact_8214_ATP_Olambda__32,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_ok(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_32
tff(fact_8215_ATP_Olambda__33,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ol(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_33
tff(fact_8216_ATP_Olambda__34,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A] : aa(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_adl(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),Uu) = aa(fun(B,fun(C,product_prod(product_prod(A,B),C))),fun(product_prod(B,C),product_prod(product_prod(A,B),C)),product_case_prod(B,C,product_prod(product_prod(A,B),C)),aTP_Lamp_adk(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).

% ATP.lambda_34
tff(fact_8217_ATP_Olambda__35,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B))] : aa(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))),aTP_Lamp_abw(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Uu) = 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_abv(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).

% ATP.lambda_35
tff(fact_8218_ATP_Olambda__36,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_nj(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = 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_ni(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_36
tff(fact_8219_ATP_Olambda__37,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_qz(real,real),Uu) = suminf(real,aTP_Lamp_da(real,fun(nat,real),Uu)) ).

% ATP.lambda_37
tff(fact_8220_ATP_Olambda__38,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(real,real)),aTP_Lamp_ads(real,filter(product_prod(real,real))),Uu) = principal(product_prod(real,real),aa(fun(product_prod(real,real),$o),set(product_prod(real,real)),collect(product_prod(real,real)),aa(fun(real,fun(real,$o)),fun(product_prod(real,real),$o),product_case_prod(real,real,$o),aTP_Lamp_adr(real,fun(real,fun(real,$o)),Uu)))) ).

% ATP.lambda_38
tff(fact_8221_ATP_Olambda__39,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(complex,complex)),aTP_Lamp_adq(real,filter(product_prod(complex,complex))),Uu) = principal(product_prod(complex,complex),aa(fun(product_prod(complex,complex),$o),set(product_prod(complex,complex)),collect(product_prod(complex,complex)),aa(fun(complex,fun(complex,$o)),fun(product_prod(complex,complex),$o),product_case_prod(complex,complex,$o),aTP_Lamp_adp(real,fun(complex,fun(complex,$o)),Uu)))) ).

% ATP.lambda_39
tff(fact_8222_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real] : aa(real,filter(product_prod(A,A)),aTP_Lamp_adn(real,filter(product_prod(A,A))),Uu) = principal(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_adm(real,fun(A,fun(A,$o)),Uu)))) ) ).

% ATP.lambda_40
tff(fact_8223_ATP_Olambda__41,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_oi(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu)) ).

% ATP.lambda_41
tff(fact_8224_ATP_Olambda__42,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_of(list(A),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu)) ).

% ATP.lambda_42
tff(fact_8225_ATP_Olambda__43,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_lu(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_43
tff(fact_8226_ATP_Olambda__44,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_lt(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).

% ATP.lambda_44
tff(fact_8227_ATP_Olambda__45,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bc(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_45
tff(fact_8228_ATP_Olambda__46,axiom,
    ! [Uu: nat] : aa(nat,extended_enat,aTP_Lamp_agr(nat,extended_enat),Uu) = extended_enat2(aa(nat,nat,suc,Uu)) ).

% ATP.lambda_46
tff(fact_8229_ATP_Olambda__47,axiom,
    ! [Uu: set(fun(nat,rat))] :
      ( aa(set(fun(nat,rat)),$o,aTP_Lamp_ail(set(fun(nat,rat)),$o),Uu)
    <=> ? [X: fun(nat,rat)] :
          ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X),X)
          & ( Uu = aa(fun(fun(nat,rat),$o),set(fun(nat,rat)),collect(fun(nat,rat)),aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X)) ) ) ) ).

% ATP.lambda_47
tff(fact_8230_ATP_Olambda__48,axiom,
    ! [Uu: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aTP_Lamp_ace(fun(nat,rat),$o),Uu)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K2: nat] :
            ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N4)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,Uu,N4)) ) ) ) ).

% ATP.lambda_48
tff(fact_8231_ATP_Olambda__49,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aTP_Lamp_acd(product_prod(A,A),$o),Uu)
    <=> ? [X: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X) ) ).

% ATP.lambda_49
tff(fact_8232_ATP_Olambda__50,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_ne(real,$o),Uu)
    <=> ? [I4: int,N4: nat] :
          ( ( Uu = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),I4)),aa(nat,real,semiring_1_of_nat(real),N4)) )
          & ( N4 != zero_zero(nat) ) ) ) ).

% ATP.lambda_50
tff(fact_8233_ATP_Olambda__51,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_nc(real,$o),Uu)
    <=> ? [I4: int,J3: int] :
          ( ( Uu = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),I4)),aa(int,real,ring_1_of_int(real),J3)) )
          & ( J3 != zero_zero(int) ) ) ) ).

% ATP.lambda_51
tff(fact_8234_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_xf(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y3: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) ) ) ) ).

% ATP.lambda_52
tff(fact_8235_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_xe(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y3: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X) ) ) ) ).

% ATP.lambda_53
tff(fact_8236_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_xc(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y3: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3) )
              & ( X != Y3 ) ) ) ) ).

% ATP.lambda_54
tff(fact_8237_ATP_Olambda__55,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_ls(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_55
tff(fact_8238_ATP_Olambda__56,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_mb(num,fun(nat,option(num)),Uu),Uua) = aa(num,option(num),aa(fun(num,option(num)),fun(num,option(num)),aa(fun(num,option(num)),fun(fun(num,option(num)),fun(num,option(num))),aa(option(num),fun(fun(num,option(num)),fun(fun(num,option(num)),fun(num,option(num)))),case_num(option(num)),aa(num,option(num),some(num),one2)),aTP_Lamp_lz(nat,fun(num,option(num)),Uua)),aTP_Lamp_ma(nat,fun(num,option(num)),Uua)),Uu) ).

% ATP.lambda_56
tff(fact_8239_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_ij(A,fun(nat,A),Uu),Uua) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_57
tff(fact_8240_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_go(nat,fun(nat,A),Uu),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(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_58
tff(fact_8241_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_ii(A,fun(nat,A),Uu),Uua) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_59
tff(fact_8242_ATP_Olambda__60,axiom,
    ! [Uu: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_el(fun(nat,real),fun(nat,real),Uu),Uua) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),zero_zero(real),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% ATP.lambda_60
tff(fact_8243_ATP_Olambda__61,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: product_prod(A,C),Uua: product_prod(C,B)] :
      aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_su(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = $ite(aa(product_prod(A,C),C,product_snd(A,C),Uu) = aa(product_prod(C,B),C,product_fst(C,B),Uua),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).

% ATP.lambda_61
tff(fact_8244_ATP_Olambda__62,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      aa(nat,extended_enat,aTP_Lamp_agp(extended_enat,fun(nat,extended_enat),Uu),Uua) = aa(extended_enat,extended_enat,
        aa(extended_enat,fun(extended_enat,extended_enat),aa(fun(nat,extended_enat),fun(extended_enat,fun(extended_enat,extended_enat)),extended_case_enat(extended_enat),aTP_Lamp_ago(nat,fun(nat,extended_enat),Uua)),
          $ite(Uua = zero_zero(nat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat))),
        Uu) ).

% ATP.lambda_62
tff(fact_8245_ATP_Olambda__63,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_agn(extended_enat,fun(nat,extended_enat),Uu),Uua) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),aa(fun(nat,extended_enat),fun(extended_enat,fun(extended_enat,extended_enat)),extended_case_enat(extended_enat),aTP_Lamp_agm(nat,fun(nat,extended_enat),Uua)),zero_zero(extended_enat)),Uu) ).

% ATP.lambda_63
tff(fact_8246_ATP_Olambda__64,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_agl(extended_enat,fun(nat,extended_enat),Uu),Uua) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),aa(fun(nat,extended_enat),fun(extended_enat,fun(extended_enat,extended_enat)),extended_case_enat(extended_enat),aTP_Lamp_agk(nat,fun(nat,extended_enat),Uua)),extend4730790105801354508finity(extended_enat)),Uu) ).

% ATP.lambda_64
tff(fact_8247_ATP_Olambda__65,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_afi(extended_enat,fun(nat,$o),Uu),Uua)
    <=> aa(extended_enat,$o,aa($o,fun(extended_enat,$o),aa(fun(nat,$o),fun($o,fun(extended_enat,$o)),extended_case_enat($o),aa(nat,fun(nat,$o),aTP_Lamp_aj(nat,fun(nat,$o)),Uua)),$false),Uu) ) ).

% ATP.lambda_65
tff(fact_8248_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_gp(nat,fun(nat,A),Uu),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(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_66
tff(fact_8249_ATP_Olambda__67,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_afj(extended_enat,fun(nat,$o),Uu),Uua)
    <=> aa(extended_enat,$o,aa($o,fun(extended_enat,$o),aa(fun(nat,$o),fun($o,fun(extended_enat,$o)),extended_case_enat($o),aa(nat,fun(nat,$o),ord_less(nat),Uua)),$true),Uu) ) ).

% ATP.lambda_67
tff(fact_8250_ATP_Olambda__68,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_zl(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),top_top(A)) ) ).

% ATP.lambda_68
tff(fact_8251_ATP_Olambda__69,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_zj(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ).

% ATP.lambda_69
tff(fact_8252_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_mc(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_mb(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_70
tff(fact_8253_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_lz(nat,fun(num,option(num)),Uu),Uua) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_lt(num,option(num))),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_71
tff(fact_8254_ATP_Olambda__72,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_lx(num,fun(nat,option(num)),Uu),Uua) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_lt(num,option(num))),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_72
tff(fact_8255_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_afq(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_73
tff(fact_8256_ATP_Olambda__74,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_oh(list(list(A)),fun(nat,list(A)),Uu),Uua) = map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_og(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu))) ).

% ATP.lambda_74
tff(fact_8257_ATP_Olambda__75,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gn(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_75
tff(fact_8258_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gl(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gk(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_76
tff(fact_8259_ATP_Olambda__77,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_cl(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_77
tff(fact_8260_ATP_Olambda__78,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ea(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_78
tff(fact_8261_ATP_Olambda__79,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_en(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))) ).

% ATP.lambda_79
tff(fact_8262_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gd(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_80
tff(fact_8263_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hj(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_81
tff(fact_8264_ATP_Olambda__82,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_il(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_82
tff(fact_8265_ATP_Olambda__83,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ik(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_83
tff(fact_8266_ATP_Olambda__84,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_ki(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
        l2: int,
        l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),code_int_of_integer(Uu)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_84
tff(fact_8267_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_cj(nat,fun(nat,a)),Uu),Uua) = $let(
        m3: a,
        m3:= aa(a,a,aa(a,fun(a,a),times_times(a),aa(num,a,numeral_numeral(a),aa(num,num,bit0,one2))),aa(nat,a,semiring_1_of_nat(a),Uu)),
        $ite(Uua = zero_zero(nat),m3,aa(a,a,aa(a,fun(a,a),plus_plus(a),m3),one_one(a))) ) ).

% ATP.lambda_85
tff(fact_8268_ATP_Olambda__86,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_iy(complex,fun(real,$o),Uu),Uua)
    <=> ( ( sgn_sgn(complex,Uu) = cis(Uua) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi) ) ) ).

% ATP.lambda_86
tff(fact_8269_ATP_Olambda__87,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_ku(real,fun(int,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_87
tff(fact_8270_ATP_Olambda__88,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_kv(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_88
tff(fact_8271_ATP_Olambda__89,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_da(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat))))) ).

% ATP.lambda_89
tff(fact_8272_ATP_Olambda__90,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ra(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% ATP.lambda_90
tff(fact_8273_ATP_Olambda__91,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_91
tff(fact_8274_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gf(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_92
tff(fact_8275_ATP_Olambda__93,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_ml(rat,fun(product_prod(int,int),$o),Uu),Uua)
    <=> ( ( Uu = fract(aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Uua))
        & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) ) ) ).

% ATP.lambda_93
tff(fact_8276_ATP_Olambda__94,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gi(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_94
tff(fact_8277_ATP_Olambda__95,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fp(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_95
tff(fact_8278_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gy(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_96
tff(fact_8279_ATP_Olambda__97,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_pi(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_97
tff(fact_8280_ATP_Olambda__98,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_lp(set(option(A)),fun(option(A),$o),Uu),Uua)
    <=> ( member(option(A),Uua,Uu)
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_98
tff(fact_8281_ATP_Olambda__99,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gc(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% ATP.lambda_99
tff(fact_8282_ATP_Olambda__100,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_qk(set(set(A)),fun(set(A),$o),Uu),Uua)
    <=> ( member(set(A),Uua,Uu)
        & ! [X: set(A)] :
            ( member(set(A),X,Uu)
           => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),X) ) ) ) ).

% ATP.lambda_100
tff(fact_8283_ATP_Olambda__101,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),aTP_Lamp_aew(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),Uu),Uua)
    <=> ( 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))),Uu),Uua)
        & ! [A6: A,B5: A,C4: 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),B5),Uua)
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),C4),Uu) )
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5),Uu) ) ) ) ).

% ATP.lambda_101
tff(fact_8284_ATP_Olambda__102,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_dv(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_102
tff(fact_8285_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_wp(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu)),exp(real,Uua)) ).

% ATP.lambda_103
tff(fact_8286_ATP_Olambda__104,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_so(set(A),fun(set(A),$o)),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uu),Uua)
        & finite_finite(A,Uua) ) ) ).

% ATP.lambda_104
tff(fact_8287_ATP_Olambda__105,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zm(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uua)),Uua) ).

% ATP.lambda_105
tff(fact_8288_ATP_Olambda__106,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fw(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uu) ).

% ATP.lambda_106
tff(fact_8289_ATP_Olambda__107,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fv(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uua) ).

% ATP.lambda_107
tff(fact_8290_ATP_Olambda__108,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_abd(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_108
tff(fact_8291_ATP_Olambda__109,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_ael(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_109
tff(fact_8292_ATP_Olambda__110,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_by(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_110
tff(fact_8293_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_al(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_111
tff(fact_8294_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ja(A,fun(A,$o),Uu),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu) ) ) ) ).

% ATP.lambda_112
tff(fact_8295_ATP_Olambda__113,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_vb(real,fun(nat,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ).

% ATP.lambda_113
tff(fact_8296_ATP_Olambda__114,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ck(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat)))) ).

% ATP.lambda_114
tff(fact_8297_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: set(A),Uua: list(A)] :
          ( aa(list(A),$o,aTP_Lamp_ahg(set(A),fun(list(A),$o),Uu),Uua)
        <=> ( sorted_wrt(A,ord_less(A),Uua)
            & ( aa(list(A),set(A),set2(A),Uua) = Uu ) ) ) ) ).

% ATP.lambda_115
tff(fact_8298_ATP_Olambda__116,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_abk(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ) ).

% ATP.lambda_116
tff(fact_8299_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_abn(nat,fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu)) ) ) ).

% ATP.lambda_117
tff(fact_8300_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_118
tff(fact_8301_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_119
tff(fact_8302_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ca(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_120
tff(fact_8303_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_121
tff(fact_8304_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_co(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_122
tff(fact_8305_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_br(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_123
tff(fact_8306_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_124
tff(fact_8307_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cc(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_125
tff(fact_8308_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_126
tff(fact_8309_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_acz(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y3: A] :
                ( aa(A,$o,Uu,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y3) ) ) ) ) ).

% ATP.lambda_127
tff(fact_8310_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_mt(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y3: A] :
                ( aa(A,$o,Uu,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Uua) ) ) ) ) ).

% ATP.lambda_128
tff(fact_8311_ATP_Olambda__129,axiom,
    ! [A: $tType,Uu: fun(nat,sum_sum(A,nat)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ahq(fun(nat,sum_sum(A,nat)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,sum_sum(A,nat),Uu,Uua) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ) ).

% ATP.lambda_129
tff(fact_8312_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afr(fun(A,A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,Uu,Uua)),Uua) ) ) ).

% ATP.lambda_130
tff(fact_8313_ATP_Olambda__131,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,product_prod(nat,A),aTP_Lamp_agt(fun(A,nat),fun(A,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(A,nat,Uu,Uua)),Uua) ).

% ATP.lambda_131
tff(fact_8314_ATP_Olambda__132,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_xo(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_132
tff(fact_8315_ATP_Olambda__133,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_fg(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_133
tff(fact_8316_ATP_Olambda__134,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_vk(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_134
tff(fact_8317_ATP_Olambda__135,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_vj(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))) ).

% ATP.lambda_135
tff(fact_8318_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_mm(nat,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uu) ).

% ATP.lambda_136
tff(fact_8319_ATP_Olambda__137,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_wf(fun(A,B),fun(A,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua))),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_137
tff(fact_8320_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_if(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_138
tff(fact_8321_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hy(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_139
tff(fact_8322_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hv(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_140
tff(fact_8323_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ia(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_141
tff(fact_8324_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_eb(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_142
tff(fact_8325_ATP_Olambda__143,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_lq(num,fun(num,int),Uu),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_143
tff(fact_8326_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ih(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua)) ) ).

% ATP.lambda_144
tff(fact_8327_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_pj(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_145
tff(fact_8328_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ib(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_146
tff(fact_8329_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ic(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_147
tff(fact_8330_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_vf(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_148
tff(fact_8331_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ve(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_149
tff(fact_8332_ATP_Olambda__150,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_dz(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_150
tff(fact_8333_ATP_Olambda__151,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_dw(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_151
tff(fact_8334_ATP_Olambda__152,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: option(product_prod(A,B))] :
      ( aa(option(product_prod(A,B)),$o,aTP_Lamp_ahv(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),Uu),Uua)
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(option(product_prod(A,B)),set(product_prod(A,B)),set_option(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),$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),Uu))) ) ).

% ATP.lambda_152
tff(fact_8335_ATP_Olambda__153,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_acf(num,fun(num,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Uu)),aa(num,int,numeral_numeral(int),Uua)) ).

% ATP.lambda_153
tff(fact_8336_ATP_Olambda__154,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_qr(list(A),fun(list(A),$o)),Uu),Uua)
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_154
tff(fact_8337_ATP_Olambda__155,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_kj(code_integer,fun(code_integer,num)),Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ).

% ATP.lambda_155
tff(fact_8338_ATP_Olambda__156,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_kk(code_integer,fun(code_integer,nat)),Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ).

% ATP.lambda_156
tff(fact_8339_ATP_Olambda__157,axiom,
    ! [A: $tType,Uu: set(A),Uua: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_aif(set(A),fun(option(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(option(A),set(A),set_option(A),Uua)),Uu) ) ).

% ATP.lambda_157
tff(fact_8340_ATP_Olambda__158,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fr(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_158
tff(fact_8341_ATP_Olambda__159,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_ng(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_159
tff(fact_8342_ATP_Olambda__160,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_ns(set(nat),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua),Uu) ) ).

% ATP.lambda_160
tff(fact_8343_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_au(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_161
tff(fact_8344_ATP_Olambda__162,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_av(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_162
tff(fact_8345_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_et(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_163
tff(fact_8346_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ge(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).

% ATP.lambda_164
tff(fact_8347_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahh(fun(A,A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),aa(A,A,Uu,Uua)) ) ) ).

% ATP.lambda_165
tff(fact_8348_ATP_Olambda__166,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_np(fun(nat,A),fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu,Uua)) ).

% ATP.lambda_166
tff(fact_8349_ATP_Olambda__167,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_aan(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).

% ATP.lambda_167
tff(fact_8350_ATP_Olambda__168,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_agi(nat,fun(vEBT_VEBT,vEBT_VEBT),Uu),Uua) = vEBT_VEBT_elim_dead(Uua,extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% ATP.lambda_168
tff(fact_8351_ATP_Olambda__169,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jh(A,fun(nat,A),Uu),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_169
tff(fact_8352_ATP_Olambda__170,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_abr(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu,Uua))) ).

% ATP.lambda_170
tff(fact_8353_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_va(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_171
tff(fact_8354_ATP_Olambda__172,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_be(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_172
tff(fact_8355_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ow(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_173
tff(fact_8356_ATP_Olambda__174,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_abm(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ).

% ATP.lambda_174
tff(fact_8357_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ur(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_175
tff(fact_8358_ATP_Olambda__176,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_oe(nat,fun(list(A),$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_176
tff(fact_8359_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fs(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_177
tff(fact_8360_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jl(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_178
tff(fact_8361_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fk(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_179
tff(fact_8362_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aj(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_8363_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aav(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_181
tff(fact_8364_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afz(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_182
tff(fact_8365_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afe(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_183
tff(fact_8366_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afl(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_184
tff(fact_8367_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_kw(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_185
tff(fact_8368_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_fu(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_186
tff(fact_8369_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_op(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_187
tff(fact_8370_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_vp(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_188
tff(fact_8371_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_io(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_8372_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aaf(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_190
tff(fact_8373_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_191
tff(fact_8374_ATP_Olambda__192,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aaw(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_192
tff(fact_8375_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_xl(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_193
tff(fact_8376_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aga(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_194
tff(fact_8377_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aff(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_195
tff(fact_8378_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afm(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_196
tff(fact_8379_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afy(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_197
tff(fact_8380_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_cp(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_198
tff(fact_8381_ATP_Olambda__199,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_up(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_199
tff(fact_8382_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_vo(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_200
tff(fact_8383_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aa(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_201
tff(fact_8384_ATP_Olambda__202,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ir(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_202
tff(fact_8385_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ji(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_203
tff(fact_8386_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_qg(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_204
tff(fact_8387_ATP_Olambda__205,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_no(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_205
tff(fact_8388_ATP_Olambda__206,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_adt(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_206
tff(fact_8389_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aag(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_207
tff(fact_8390_ATP_Olambda__208,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_in(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_208
tff(fact_8391_ATP_Olambda__209,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_qj(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_209
tff(fact_8392_ATP_Olambda__210,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ay(nat,fun(nat,$o),Uu),Uua)
    <=> dvd_dvd(nat,Uua,Uu) ) ).

% ATP.lambda_210
tff(fact_8393_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_aw(A,fun(A,$o),Uu),Uua)
        <=> dvd_dvd(A,Uua,Uu) ) ) ).

% ATP.lambda_211
tff(fact_8394_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_le(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uu) ).

% ATP.lambda_212
tff(fact_8395_ATP_Olambda__213,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_nl(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_213
tff(fact_8396_ATP_Olambda__214,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_nq(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),Uu) ).

% ATP.lambda_214
tff(fact_8397_ATP_Olambda__215,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_aam(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).

% ATP.lambda_215
tff(fact_8398_ATP_Olambda__216,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_aeh(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu) ).

% ATP.lambda_216
tff(fact_8399_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : aa(A,A,aTP_Lamp_aak(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ).

% ATP.lambda_217
tff(fact_8400_ATP_Olambda__218,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_uu(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_218
tff(fact_8401_ATP_Olambda__219,axiom,
    ! [Uu: set(extended_enat),Uua: extended_enat] :
      ( aa(extended_enat,$o,aTP_Lamp_agj(set(extended_enat),fun(extended_enat,$o),Uu),Uua)
    <=> member(extended_enat,Uua,Uu) ) ).

% ATP.lambda_219
tff(fact_8402_ATP_Olambda__220,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_acy(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_220
tff(fact_8403_ATP_Olambda__221,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_a(set(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,Uu) ) ).

% ATP.lambda_221
tff(fact_8404_ATP_Olambda__222,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_abj(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Uua),Uu) ).

% ATP.lambda_222
tff(fact_8405_ATP_Olambda__223,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_od(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_223
tff(fact_8406_ATP_Olambda__224,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_wb(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_224
tff(fact_8407_ATP_Olambda__225,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_wm(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_225
tff(fact_8408_ATP_Olambda__226,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aeu(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_226
tff(fact_8409_ATP_Olambda__227,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_cz(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))) ).

% ATP.lambda_227
tff(fact_8410_ATP_Olambda__228,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_cy(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)) ).

% ATP.lambda_228
tff(fact_8411_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_xb(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Uua)) ) ).

% ATP.lambda_229
tff(fact_8412_ATP_Olambda__230,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_acv(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_230
tff(fact_8413_ATP_Olambda__231,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ee(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_231
tff(fact_8414_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ef(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_232
tff(fact_8415_ATP_Olambda__233,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_233
tff(fact_8416_ATP_Olambda__234,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_234
tff(fact_8417_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_mi(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),ord_min(A),Uu)),Uua)) ) ).

% ATP.lambda_235
tff(fact_8418_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_md(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),ord_max(A),Uu)),Uua)) ) ).

% ATP.lambda_236
tff(fact_8419_ATP_Olambda__237,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_abf(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),sup_sup(A),Uu)),Uua)) ) ).

% ATP.lambda_237
tff(fact_8420_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_ym(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),inf_inf(A),Uu)),Uua)) ) ).

% ATP.lambda_238
tff(fact_8421_ATP_Olambda__239,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_ma(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_239
tff(fact_8422_ATP_Olambda__240,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_ly(num,fun(nat,option(num)),Uu),Uua) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_240
tff(fact_8423_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_acr(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_acq(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_241
tff(fact_8424_ATP_Olambda__242,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_lm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_ll(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_242
tff(fact_8425_ATP_Olambda__243,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_lk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lj(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_243
tff(fact_8426_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_li(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_lh(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_244
tff(fact_8427_ATP_Olambda__245,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_lg(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_lf(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_245
tff(fact_8428_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lc(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_246
tff(fact_8429_ATP_Olambda__247,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_re(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_rd(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_247
tff(fact_8430_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_qh(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_248
tff(fact_8431_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A] : aa(A,set(A),aTP_Lamp_xj(real,fun(A,set(A)),Uu),Uua) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_xi(real,fun(A,fun(A,$o)),Uu),Uua)) ) ).

% ATP.lambda_249
tff(fact_8432_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_iw(nat,fun(nat,complex),Uu),Uua) = cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua))),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_250
tff(fact_8433_ATP_Olambda__251,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_rq(A,fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Uu)),Uua)),aa(A,A,inverse_inverse(A),Uu))) ) ).

% ATP.lambda_251
tff(fact_8434_ATP_Olambda__252,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aax(fun(B,option(A)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(A),Uu,Uua) != none(A) ) ) ).

% ATP.lambda_252
tff(fact_8435_ATP_Olambda__253,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_abq(fun(A,option(B)),fun(A,$o),Uu),Uua)
    <=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).

% ATP.lambda_253
tff(fact_8436_ATP_Olambda__254,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_aem(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image(B,set(product_prod(A,B)),aTP_Lamp_ael(A,fun(B,set(product_prod(A,B))),Uua),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_254
tff(fact_8437_ATP_Olambda__255,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_sv(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_su(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua),Uu)) ).

% ATP.lambda_255
tff(fact_8438_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_hw(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_256
tff(fact_8439_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_bd(nat,fun(nat,$o),Uu),Uua)
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))) ) ).

% ATP.lambda_257
tff(fact_8440_ATP_Olambda__258,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ig(A,fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua))) ) ).

% ATP.lambda_258
tff(fact_8441_ATP_Olambda__259,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_id(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_259
tff(fact_8442_ATP_Olambda__260,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_ie(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_260
tff(fact_8443_ATP_Olambda__261,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ft(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_261
tff(fact_8444_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_eu(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_262
tff(fact_8445_ATP_Olambda__263,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aek(nat,fun(nat,set(nat)),Uu),Uua) = set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua)) ).

% ATP.lambda_263
tff(fact_8446_ATP_Olambda__264,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_uz(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_264
tff(fact_8447_ATP_Olambda__265,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_qd(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_265
tff(fact_8448_ATP_Olambda__266,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_adk(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = aa(product_prod(A,B),fun(C,product_prod(product_prod(A,B),C)),product_Pair(product_prod(A,B),C),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_266
tff(fact_8449_ATP_Olambda__267,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_ago(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uua)) ).

% ATP.lambda_267
tff(fact_8450_ATP_Olambda__268,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_agm(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua)) ).

% ATP.lambda_268
tff(fact_8451_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_agk(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)) ).

% ATP.lambda_269
tff(fact_8452_ATP_Olambda__270,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_adv(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uu,Uua)) ) ).

% ATP.lambda_270
tff(fact_8453_ATP_Olambda__271,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_adu(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uua,Uu)) ) ).

% ATP.lambda_271
tff(fact_8454_ATP_Olambda__272,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_abe(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).

% ATP.lambda_272
tff(fact_8455_ATP_Olambda__273,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_aen(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_273
tff(fact_8456_ATP_Olambda__274,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mj(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uu),Uua)) ).

% ATP.lambda_274
tff(fact_8457_ATP_Olambda__275,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mk(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uua),Uu)) ).

% ATP.lambda_275
tff(fact_8458_ATP_Olambda__276,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kp(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu),Uua)) ).

% ATP.lambda_276
tff(fact_8459_ATP_Olambda__277,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kq(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu)) ).

% ATP.lambda_277
tff(fact_8460_ATP_Olambda__278,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_afp(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8461_ATP_Olambda__279,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_ahi(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_gfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_279
tff(fact_8462_ATP_Olambda__280,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_dr(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_280
tff(fact_8463_ATP_Olambda__281,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_gt(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_281
tff(fact_8464_ATP_Olambda__282,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ut(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_282
tff(fact_8465_ATP_Olambda__283,axiom,
    ! [Uu: fun(nat,rat),Uua: nat] : aa(nat,rat,aTP_Lamp_ox(fun(nat,rat),fun(nat,rat),Uu),Uua) = aa(rat,rat,inverse_inverse(rat),aa(nat,rat,Uu,Uua)) ).

% ATP.lambda_283
tff(fact_8466_ATP_Olambda__284,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_rr(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_284
tff(fact_8467_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qb(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_285
tff(fact_8468_ATP_Olambda__286,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_wo(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_286
tff(fact_8469_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8470_ATP_Olambda__288,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_fq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_288
tff(fact_8471_ATP_Olambda__289,axiom,
    ! [Uu: fun(nat,rat),Uua: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_or(fun(nat,rat),fun(nat,rat)),Uu),Uua) = aa(rat,rat,uminus_uminus(rat),aa(nat,rat,Uu,Uua)) ).

% ATP.lambda_289
tff(fact_8472_ATP_Olambda__290,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fb(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_8473_ATP_Olambda__291,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fn(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_291
tff(fact_8474_ATP_Olambda__292,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_292
tff(fact_8475_ATP_Olambda__293,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sh(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_293
tff(fact_8476_ATP_Olambda__294,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_294
tff(fact_8477_ATP_Olambda__295,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aai(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_8478_ATP_Olambda__296,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_296
tff(fact_8479_ATP_Olambda__297,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sl(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_297
tff(fact_8480_ATP_Olambda__298,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ds(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_298
tff(fact_8481_ATP_Olambda__299,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_bi(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_8482_ATP_Olambda__300,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qq(fun(A,A),fun(A,A),Uu),Uua) = tanh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_300
tff(fact_8483_ATP_Olambda__301,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pu(fun(A,A),fun(A,A),Uu),Uua) = sinh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_8484_ATP_Olambda__302,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pv(fun(A,A),fun(A,A),Uu),Uua) = cosh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_302
tff(fact_8485_ATP_Olambda__303,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_303
tff(fact_8486_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pq(fun(A,A),fun(A,A),Uu),Uua) = sin(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_304
tff(fact_8487_ATP_Olambda__305,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pn(fun(A,A),fun(A,A),Uu),Uua) = exp(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_305
tff(fact_8488_ATP_Olambda__306,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_vi(fun(nat,real),fun(nat,real),Uu),Uua) = cos(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_306
tff(fact_8489_ATP_Olambda__307,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qc(fun(A,A),fun(A,A),Uu),Uua) = cos(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_307
tff(fact_8490_ATP_Olambda__308,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_mg(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_308
tff(fact_8491_ATP_Olambda__309,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_abp(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_309
tff(fact_8492_ATP_Olambda__310,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_na(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uua)) ).

% ATP.lambda_310
tff(fact_8493_ATP_Olambda__311,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_aaj(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uua)) ).

% ATP.lambda_311
tff(fact_8494_ATP_Olambda__312,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: fun(A,extended_enat),Uua: A] : aa(A,extended_enat,aTP_Lamp_agq(fun(A,extended_enat),fun(A,extended_enat),Uu),Uua) = aa(extended_enat,extended_enat,extended_eSuc,aa(A,extended_enat,Uu,Uua)) ) ).

% ATP.lambda_312
tff(fact_8495_ATP_Olambda__313,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_pg(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_313
tff(fact_8496_ATP_Olambda__314,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_ui(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_314
tff(fact_8497_ATP_Olambda__315,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_315
tff(fact_8498_ATP_Olambda__316,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_316
tff(fact_8499_ATP_Olambda__317,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yx(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_317
tff(fact_8500_ATP_Olambda__318,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ).

% ATP.lambda_318
tff(fact_8501_ATP_Olambda__319,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_jx(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_319
tff(fact_8502_ATP_Olambda__320,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_nw(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_320
tff(fact_8503_ATP_Olambda__321,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aes(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ak(nat,fun(nat,$o)),Uu)) ).

% ATP.lambda_321
tff(fact_8504_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real] : aa(real,filter(A),aTP_Lamp_adx(A,fun(real,filter(A)),Uu),Uua) = principal(A,aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_adw(A,fun(real,fun(A,$o)),Uu),Uua))) ) ).

% ATP.lambda_322
tff(fact_8505_ATP_Olambda__323,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_mo(list(A),fun(A,$o),Uu),Uua)
    <=> ? [I4: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu)) ) ) ).

% ATP.lambda_323
tff(fact_8506_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_qo(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image(set(A),A,F5,Uu) )
              & ! [X: set(A)] :
                  ( member(set(A),X,Uu)
                 => member(A,aa(set(A),A,F5,X),X) ) ) ) ) ).

% ATP.lambda_324
tff(fact_8507_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_qp(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image(set(A),A,F5,Uu) )
              & ! [X: set(A)] :
                  ( member(set(A),X,Uu)
                 => member(A,aa(set(A),A,F5,X),X) ) ) ) ) ).

% ATP.lambda_325
tff(fact_8508_ATP_Olambda__326,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_rb(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image(set(A),A,F5,Uu) )
              & ! [X: set(A)] :
                  ( member(set(A),X,Uu)
                 => member(A,aa(set(A),A,F5,X),X) ) ) ) ) ).

% ATP.lambda_326
tff(fact_8509_ATP_Olambda__327,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afx(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( member(A,X,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X) ) ) ) ).

% ATP.lambda_327
tff(fact_8510_ATP_Olambda__328,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_pk(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( member(A,X,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X) ) ) ) ).

% ATP.lambda_328
tff(fact_8511_ATP_Olambda__329,axiom,
    ! [Uu: set(real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_acw(set(real),fun(real,$o),Uu),Uua)
    <=> ! [X: real] :
          ( member(real,X,Uu)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Uua) ) ) ).

% ATP.lambda_329
tff(fact_8512_ATP_Olambda__330,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afw(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( member(A,X,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Uua) ) ) ) ).

% ATP.lambda_330
tff(fact_8513_ATP_Olambda__331,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_pl(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( member(A,X,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Uua) ) ) ) ).

% ATP.lambda_331
tff(fact_8514_ATP_Olambda__332,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_wq(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y3)
             => aa(A,$o,Uu,Y3) ) ) ) ).

% ATP.lambda_332
tff(fact_8515_ATP_Olambda__333,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_agg(set(product_prod(A,A)),fun(set(A),$o),Uu),Uua)
    <=> ! [X: A] :
          ( member(A,X,Uua)
         => ! [Xa2: A] :
              ( member(A,Xa2,Uua)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa2),Uu)
                | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X),Uu) ) ) ) ) ).

% ATP.lambda_333
tff(fact_8516_ATP_Olambda__334,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ahp(set(product_prod(A,B)),fun(A,$o),Uu),Uua)
    <=> ? [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),Uua),Y3),Uu) ) ).

% ATP.lambda_334
tff(fact_8517_ATP_Olambda__335,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ny(fun(B,option(A)),fun(A,$o),Uu),Uua)
    <=> ? [A6: B] : aa(B,option(A),Uu,A6) = aa(A,option(A),some(A),Uua) ) ).

% ATP.lambda_335
tff(fact_8518_ATP_Olambda__336,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_abs(fun(A,option(B)),fun(product_prod(A,B),$o),Uu),Uua)
    <=> ? [A6: A,B5: B] :
          ( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
          & ( aa(A,option(B),Uu,A6) = aa(B,option(B),some(B),B5) ) ) ) ).

% ATP.lambda_336
tff(fact_8519_ATP_Olambda__337,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
      ( aa(product_prod(set(A),set(A)),$o,aTP_Lamp_act(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),Uu),Uua)
    <=> ? [X7: set(A),Y9: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X7),Y9) )
          & ( X7 != bot_bot(set(A)) )
          & ! [X: A] :
              ( member(A,X,Y9)
             => ? [Xa2: A] :
                  ( member(A,Xa2,X7)
                  & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X),Uu) ) ) ) ) ).

% ATP.lambda_337
tff(fact_8520_ATP_Olambda__338,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,rat,aTP_Lamp_oy(nat,fun(nat,rat),Uu),Uua) = aa(nat,rat,semiring_1_of_nat(rat),Uu) ).

% ATP.lambda_338
tff(fact_8521_ATP_Olambda__339,axiom,
    ! [Uu: int,Uua: nat] : aa(nat,rat,aTP_Lamp_pc(int,fun(nat,rat),Uu),Uua) = aa(int,rat,ring_1_of_int(rat),Uu) ).

% ATP.lambda_339
tff(fact_8522_ATP_Olambda__340,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_afg(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_340
tff(fact_8523_ATP_Olambda__341,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_afn(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_341
tff(fact_8524_ATP_Olambda__342,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: option(A),Uub: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aTP_Lamp_aie(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),Uu),Uua),Uub)
    <=> aa(option(A),$o,aa(fun(A,$o),fun(option(A),$o),aa($o,fun(fun(A,$o),fun(option(A),$o)),case_option($o,A),aa(option(B),$o,aa(fun(B,$o),fun(option(B),$o),aa($o,fun(fun(B,$o),fun(option(B),$o)),case_option($o,B),$true),aTP_Lamp_aic(B,$o)),Uub)),aa(option(B),fun(A,$o),aTP_Lamp_aid(fun(A,fun(B,$o)),fun(option(B),fun(A,$o)),Uu),Uub)),Uua) ) ).

% ATP.lambda_342
tff(fact_8525_ATP_Olambda__343,axiom,
    ! [Uu: 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)),Uu),Uua),Uub) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub),aa(nat,real,Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),one_one(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% ATP.lambda_343
tff(fact_8526_ATP_Olambda__344,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_cx(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_344
tff(fact_8527_ATP_Olambda__345,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_jq(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_345
tff(fact_8528_ATP_Olambda__346,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_az(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_346
tff(fact_8529_ATP_Olambda__347,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ba(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_347
tff(fact_8530_ATP_Olambda__348,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: num,Uua: A,Uub: A] :
          aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_bb(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),Uub)) ) ).

% ATP.lambda_348
tff(fact_8531_ATP_Olambda__349,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_yf(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_349
tff(fact_8532_ATP_Olambda__350,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ew(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_350
tff(fact_8533_ATP_Olambda__351,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ex(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_351
tff(fact_8534_ATP_Olambda__352,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ks(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_352
tff(fact_8535_ATP_Olambda__353,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_kr(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).

% ATP.lambda_353
tff(fact_8536_ATP_Olambda__354,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_fd(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_354
tff(fact_8537_ATP_Olambda__355,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_oc(fun(B,A),fun(fun(B,$o),fun(B,option(A))),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_355
tff(fact_8538_ATP_Olambda__356,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A,Uub: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(A,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_aeo(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_aen(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_356
tff(fact_8539_ATP_Olambda__357,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B,Uub: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(B,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_abi(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_abe(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_357
tff(fact_8540_ATP_Olambda__358,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_ahu(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),aa(A,option(A),some(A),Uub)),aa(A,fun(A,option(A)),aTP_Lamp_aht(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uub)),Uua) ).

% ATP.lambda_358
tff(fact_8541_ATP_Olambda__359,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] : aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_abv(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub) = fun_upd(A,option(B),Uu,Uua,aa(B,option(B),some(B),Uub)) ).

% ATP.lambda_359
tff(fact_8542_ATP_Olambda__360,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aa(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_ahj(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),Uu),Uua),Uub) = fun_upd(A,option(B),Uub,Uu,aa(B,option(B),some(B),Uua)) ).

% ATP.lambda_360
tff(fact_8543_ATP_Olambda__361,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aao(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uub),Uu),Uua) ).

% ATP.lambda_361
tff(fact_8544_ATP_Olambda__362,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: option(B),Uub: A] :
      ( aa(A,$o,aa(option(B),fun(A,$o),aTP_Lamp_aid(fun(A,fun(B,$o)),fun(option(B),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(option(B),$o,aa(fun(B,$o),fun(option(B),$o),aa($o,fun(fun(B,$o),fun(option(B),$o)),case_option($o,B),$false),aa(A,fun(B,$o),Uu,Uub)),Uua) ) ).

% ATP.lambda_362
tff(fact_8545_ATP_Olambda__363,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_363
tff(fact_8546_ATP_Olambda__364,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gk(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_364
tff(fact_8547_ATP_Olambda__365,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_tr(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_365
tff(fact_8548_ATP_Olambda__366,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_rz(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_366
tff(fact_8549_ATP_Olambda__367,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hu(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ht(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_367
tff(fact_8550_ATP_Olambda__368,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hs(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hr(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_368
tff(fact_8551_ATP_Olambda__369,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hq(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hp(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_369
tff(fact_8552_ATP_Olambda__370,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gv(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_gu(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_370
tff(fact_8553_ATP_Olambda__371,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: A,Uub: nat] :
          aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eh(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu,one_one(A),zero_zero(A))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_371
tff(fact_8554_ATP_Olambda__372,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_kl(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_372
tff(fact_8555_ATP_Olambda__373,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A)),Uub: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_ahl(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),Uu),Uua),Uub)
    <=> ( aa(set(set(A)),$o,pred_chain(set(A),Uu,ord_less(set(A))),Uub)
        & aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less(set(set(A))),Uua),Uub) ) ) ).

% ATP.lambda_373
tff(fact_8556_ATP_Olambda__374,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_zf(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_dj(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_374
tff(fact_8557_ATP_Olambda__375,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_kz(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_ky(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_375
tff(fact_8558_ATP_Olambda__376,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_sa(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_rz(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_376
tff(fact_8559_ATP_Olambda__377,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qs(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_377
tff(fact_8560_ATP_Olambda__378,axiom,
    ! [Uu: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qt(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_378
tff(fact_8561_ATP_Olambda__379,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_dx(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_379
tff(fact_8562_ATP_Olambda__380,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_gq(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_380
tff(fact_8563_ATP_Olambda__381,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_uf(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_381
tff(fact_8564_ATP_Olambda__382,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_uc(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_382
tff(fact_8565_ATP_Olambda__383,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_383
tff(fact_8566_ATP_Olambda__384,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_te(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_384
tff(fact_8567_ATP_Olambda__385,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ud(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_385
tff(fact_8568_ATP_Olambda__386,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_rc(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_386
tff(fact_8569_ATP_Olambda__387,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_dy(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,Uua,Uub)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_387
tff(fact_8570_ATP_Olambda__388,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_mq(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_388
tff(fact_8571_ATP_Olambda__389,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_389
tff(fact_8572_ATP_Olambda__390,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_mh(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub),lex(A,Uu)) ) ) ) ).

% ATP.lambda_390
tff(fact_8573_ATP_Olambda__391,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_ada(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(nat,nat,suc,Uu) )
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).

% ATP.lambda_391
tff(fact_8574_ATP_Olambda__392,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_ss(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X: A,Y3: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uua = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs5)) )
            & ( Uub = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),Uu) ) ) ) ).

% ATP.lambda_392
tff(fact_8575_ATP_Olambda__393,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_mr(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_393
tff(fact_8576_ATP_Olambda__394,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_kh(nat,fun(set(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_394
tff(fact_8577_ATP_Olambda__395,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_kg(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_395
tff(fact_8578_ATP_Olambda__396,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_aeg(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_396
tff(fact_8579_ATP_Olambda__397,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_jp(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_397
tff(fact_8580_ATP_Olambda__398,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ai(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_398
tff(fact_8581_ATP_Olambda__399,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ah(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_399
tff(fact_8582_ATP_Olambda__400,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_mp(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_400
tff(fact_8583_ATP_Olambda__401,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ql(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_401
tff(fact_8584_ATP_Olambda__402,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jv(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_402
tff(fact_8585_ATP_Olambda__403,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_ob(set(nat),fun(nat,fun(product_prod(A,nat),$o)),Uu),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),Uu) ) ).

% ATP.lambda_403
tff(fact_8586_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_nm(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))
        & member(nat,Uub,Uua) ) ) ).

% ATP.lambda_404
tff(fact_8587_ATP_Olambda__405,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_nz(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),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,Uu,aa(nat,A,nth(A,Uua),Uub)) ) ) ).

% ATP.lambda_405
tff(fact_8588_ATP_Olambda__406,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_oa(list(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uu))
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_406
tff(fact_8589_ATP_Olambda__407,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(A,fun(product_prod(A,B),$o),aTP_Lamp_abt(fun(A,option(B)),fun(A,fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ( member(product_prod(A,B),Uub,graph(A,B,Uu))
        & ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).

% ATP.lambda_407
tff(fact_8590_ATP_Olambda__408,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jk(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_408
tff(fact_8591_ATP_Olambda__409,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aca(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu))
        & ! [X: A] :
            ( member(A,X,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),X),Uu) ) ) ) ).

% ATP.lambda_409
tff(fact_8592_ATP_Olambda__410,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_acc(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu))
        & ! [X: A] :
            ( member(A,X,Uua)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Uub),Uu) ) ) ) ).

% ATP.lambda_410
tff(fact_8593_ATP_Olambda__411,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_acb(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu))
        & ! [X: A] :
            ( member(A,X,Uua)
           => ( ( Uub != X )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X),Uu) ) ) ) ) ).

% ATP.lambda_411
tff(fact_8594_ATP_Olambda__412,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_afh(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu))
        & ! [X: A] :
            ( member(A,X,Uua)
           => ( ( Uub != X )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Uub),Uu) ) ) ) ) ).

% ATP.lambda_412
tff(fact_8595_ATP_Olambda__413,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_sp(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)),Uua) ) ).

% ATP.lambda_413
tff(fact_8596_ATP_Olambda__414,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_abz(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uub) ) ) ).

% ATP.lambda_414
tff(fact_8597_ATP_Olambda__415,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_acu(set(A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,Uu,Uua)),Uub) ) ) ) ).

% ATP.lambda_415
tff(fact_8598_ATP_Olambda__416,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_afk(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( order_ofilter(A,Uu,Uua)
        & ( Uua != field2(A,Uu) )
        & order_ofilter(A,Uu,Uub)
        & ( Uub != field2(A,Uu) )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),Uub) ) ) ).

% ATP.lambda_416
tff(fact_8599_ATP_Olambda__417,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_acp(set(nat),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & member(nat,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aco(set(nat),fun(nat,fun(nat,$o)),Uu),Uub))),Uua) ) ) ).

% ATP.lambda_417
tff(fact_8600_ATP_Olambda__418,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jw(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_418
tff(fact_8601_ATP_Olambda__419,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eo(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_419
tff(fact_8602_ATP_Olambda__420,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_420
tff(fact_8603_ATP_Olambda__421,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_es(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_421
tff(fact_8604_ATP_Olambda__422,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fx(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub)) ).

% ATP.lambda_422
tff(fact_8605_ATP_Olambda__423,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_bf(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_423
tff(fact_8606_ATP_Olambda__424,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_bg(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_424
tff(fact_8607_ATP_Olambda__425,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ao(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_425
tff(fact_8608_ATP_Olambda__426,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_an(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_426
tff(fact_8609_ATP_Olambda__427,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_am(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_427
tff(fact_8610_ATP_Olambda__428,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_ack(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu) ) ) ).

% ATP.lambda_428
tff(fact_8611_ATP_Olambda__429,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ad(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uub) ) ) ).

% ATP.lambda_429
tff(fact_8612_ATP_Olambda__430,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ac(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_430
tff(fact_8613_ATP_Olambda__431,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_kx(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( dvd_dvd(nat,Uub,Uua)
        & dvd_dvd(nat,Uub,Uu) ) ) ).

% ATP.lambda_431
tff(fact_8614_ATP_Olambda__432,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aex(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uua,Uu)
        & member(A,Uub,Uu) ) ) ).

% ATP.lambda_432
tff(fact_8615_ATP_Olambda__433,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aco(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_433
tff(fact_8616_ATP_Olambda__434,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: A] :
          ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_aaa(set(A),fun(fun(A,real),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_434
tff(fact_8617_ATP_Olambda__435,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_mu(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu),Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_435
tff(fact_8618_ATP_Olambda__436,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_nd(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_436
tff(fact_8619_ATP_Olambda__437,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_fc(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_437
tff(fact_8620_ATP_Olambda__438,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_as(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_438
tff(fact_8621_ATP_Olambda__439,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aq(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_439
tff(fact_8622_ATP_Olambda__440,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ln(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_440
tff(fact_8623_ATP_Olambda__441,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_jr(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ~ dvd_dvd(B,aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_441
tff(fact_8624_ATP_Olambda__442,axiom,
    ! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_nr(list(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_442
tff(fact_8625_ATP_Olambda__443,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_iq(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_443
tff(fact_8626_ATP_Olambda__444,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_gj(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_444
tff(fact_8627_ATP_Olambda__445,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_xd(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_445
tff(fact_8628_ATP_Olambda__446,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_adp(real,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(complex,Uua,Uub)),Uu) ) ).

% ATP.lambda_446
tff(fact_8629_ATP_Olambda__447,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_adr(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(real,Uua,Uub)),Uu) ) ).

% ATP.lambda_447
tff(fact_8630_ATP_Olambda__448,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_adm(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_448
tff(fact_8631_ATP_Olambda__449,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xi(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_449
tff(fact_8632_ATP_Olambda__450,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_adw(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uub,Uu)),Uua) ) ) ).

% ATP.lambda_450
tff(fact_8633_ATP_Olambda__451,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_gs(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_451
tff(fact_8634_ATP_Olambda__452,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_iv(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_452
tff(fact_8635_ATP_Olambda__453,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_it(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_453
tff(fact_8636_ATP_Olambda__454,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_iu(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_454
tff(fact_8637_ATP_Olambda__455,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_is(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_455
tff(fact_8638_ATP_Olambda__456,axiom,
    ! [B: $tType,C: $tType,Uu: set(product_prod(C,B)),Uua: C,Uub: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_aih(set(product_prod(C,B)),fun(C,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uua),Uub),Uu) ) ).

% ATP.lambda_456
tff(fact_8639_ATP_Olambda__457,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_aii(set(product_prod(B,C)),fun(B,fun(C,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub),Uu) ) ).

% ATP.lambda_457
tff(fact_8640_ATP_Olambda__458,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_age(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub),Uu) ) ).

% ATP.lambda_458
tff(fact_8641_ATP_Olambda__459,axiom,
    ! [C: $tType,A: $tType,Uu: set(product_prod(A,C)),Uua: A,Uub: C] :
      ( aa(C,$o,aa(A,fun(C,$o),aTP_Lamp_aig(set(product_prod(A,C)),fun(A,fun(C,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uua),Uub),Uu) ) ).

% ATP.lambda_459
tff(fact_8642_ATP_Olambda__460,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_lb(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu),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),Uu) ) ).

% ATP.lambda_460
tff(fact_8643_ATP_Olambda__461,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),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),Uu) ) ).

% ATP.lambda_461
tff(fact_8644_ATP_Olambda__462,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_agf(set(product_prod(B,A)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uub),Uua),Uu) ) ).

% ATP.lambda_462
tff(fact_8645_ATP_Olambda__463,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afo(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),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),Uub),Uua),Uu) ) ).

% ATP.lambda_463
tff(fact_8646_ATP_Olambda__464,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_og(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu),Uub)),Uua) ).

% ATP.lambda_464
tff(fact_8647_ATP_Olambda__465,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_ap(nat,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu) = Uua ) ) ).

% ATP.lambda_465
tff(fact_8648_ATP_Olambda__466,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_jb(complex,fun(nat,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu ) ) ).

% ATP.lambda_466
tff(fact_8649_ATP_Olambda__467,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_iz(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_467
tff(fact_8650_ATP_Olambda__468,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dn(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_468
tff(fact_8651_ATP_Olambda__469,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_469
tff(fact_8652_ATP_Olambda__470,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_vh(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),aa(real,real,aa(real,fun(real,real),times_times(real),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_470
tff(fact_8653_ATP_Olambda__471,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_rd(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_471
tff(fact_8654_ATP_Olambda__472,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_dt(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_472
tff(fact_8655_ATP_Olambda__473,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gz(fun(nat,nat),fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),Uub)) ).

% ATP.lambda_473
tff(fact_8656_ATP_Olambda__474,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_yp(fun(nat,B),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uub)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),Uub)) ) ).

% ATP.lambda_474
tff(fact_8657_ATP_Olambda__475,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_475
tff(fact_8658_ATP_Olambda__476,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dj(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_476
tff(fact_8659_ATP_Olambda__477,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_477
tff(fact_8660_ATP_Olambda__478,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_478
tff(fact_8661_ATP_Olambda__479,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_479
tff(fact_8662_ATP_Olambda__480,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_480
tff(fact_8663_ATP_Olambda__481,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_af(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_481
tff(fact_8664_ATP_Olambda__482,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,real),Uua: fun(real,A),Uub: real] : aa(real,A,aa(fun(real,A),fun(real,A),aTP_Lamp_agv(fun(real,real),fun(fun(real,A),fun(real,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,Uu,Uub)),aa(real,A,Uua,Uub)) ) ).

% ATP.lambda_482
tff(fact_8665_ATP_Olambda__483,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rh(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_483
tff(fact_8666_ATP_Olambda__484,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_nx(fun(B,A),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_484
tff(fact_8667_ATP_Olambda__485,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_vq(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_485
tff(fact_8668_ATP_Olambda__486,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ws(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_486
tff(fact_8669_ATP_Olambda__487,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_vr(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_487
tff(fact_8670_ATP_Olambda__488,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ez(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_488
tff(fact_8671_ATP_Olambda__489,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_489
tff(fact_8672_ATP_Olambda__490,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ro(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_490
tff(fact_8673_ATP_Olambda__491,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_491
tff(fact_8674_ATP_Olambda__492,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pr(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_492
tff(fact_8675_ATP_Olambda__493,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_493
tff(fact_8676_ATP_Olambda__494,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wn(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_494
tff(fact_8677_ATP_Olambda__495,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wy(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_495
tff(fact_8678_ATP_Olambda__496,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_496
tff(fact_8679_ATP_Olambda__497,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_zy(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_497
tff(fact_8680_ATP_Olambda__498,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(real,A),Uua: fun(real,A),Uub: real] : aa(real,A,aa(fun(real,A),fun(real,A),aTP_Lamp_ahb(fun(real,A),fun(fun(real,A),fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,Uu,Uub)),aa(real,A,Uua,Uub)) ) ).

% ATP.lambda_498
tff(fact_8681_ATP_Olambda__499,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_os(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Uu),Uua),Uub) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(nat,rat,Uu,Uub)),aa(nat,rat,Uua,Uub)) ).

% ATP.lambda_499
tff(fact_8682_ATP_Olambda__500,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_jo(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_500
tff(fact_8683_ATP_Olambda__501,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_501
tff(fact_8684_ATP_Olambda__502,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_502
tff(fact_8685_ATP_Olambda__503,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_503
tff(fact_8686_ATP_Olambda__504,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ey(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_504
tff(fact_8687_ATP_Olambda__505,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_505
tff(fact_8688_ATP_Olambda__506,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_506
tff(fact_8689_ATP_Olambda__507,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_507
tff(fact_8690_ATP_Olambda__508,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pw(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_508
tff(fact_8691_ATP_Olambda__509,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_509
tff(fact_8692_ATP_Olambda__510,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_510
tff(fact_8693_ATP_Olambda__511,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wj(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_511
tff(fact_8694_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tk(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_512
tff(fact_8695_ATP_Olambda__513,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_513
tff(fact_8696_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ti(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_514
tff(fact_8697_ATP_Olambda__515,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tj(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_515
tff(fact_8698_ATP_Olambda__516,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lo(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_516
tff(fact_8699_ATP_Olambda__517,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wk(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu,Uub)) ).

% ATP.lambda_517
tff(fact_8700_ATP_Olambda__518,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_518
tff(fact_8701_ATP_Olambda__519,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),Uu),Uua),Uub) = aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,Uu,Uub)),aa(nat,rat,Uua,Uub)) ).

% ATP.lambda_519
tff(fact_8702_ATP_Olambda__520,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ei(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_520
tff(fact_8703_ATP_Olambda__521,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_cq(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_521
tff(fact_8704_ATP_Olambda__522,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_bt(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_522
tff(fact_8705_ATP_Olambda__523,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_zu(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_523
tff(fact_8706_ATP_Olambda__524,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_yy(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_524
tff(fact_8707_ATP_Olambda__525,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_tv(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_525
tff(fact_8708_ATP_Olambda__526,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,A),Uua: fun(real,A),Uub: real] : aa(real,A,aa(fun(real,A),fun(real,A),aTP_Lamp_agz(fun(real,A),fun(fun(real,A),fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,Uu,Uub)),aa(real,A,Uua,Uub)) ) ).

% ATP.lambda_526
tff(fact_8709_ATP_Olambda__527,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ou(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Uu),Uua),Uub) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Uu,Uub)),aa(nat,rat,Uua,Uub)) ).

% ATP.lambda_527
tff(fact_8710_ATP_Olambda__528,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_8711_ATP_Olambda__529,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afc(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_529
tff(fact_8712_ATP_Olambda__530,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bn(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_530
tff(fact_8713_ATP_Olambda__531,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rj(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8714_ATP_Olambda__532,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zs(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8715_ATP_Olambda__533,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pz(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_533
tff(fact_8716_ATP_Olambda__534,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_534
tff(fact_8717_ATP_Olambda__535,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ww(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_535
tff(fact_8718_ATP_Olambda__536,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_536
tff(fact_8719_ATP_Olambda__537,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oo(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_537
tff(fact_8720_ATP_Olambda__538,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_qn(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_538
tff(fact_8721_ATP_Olambda__539,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sd(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_539
tff(fact_8722_ATP_Olambda__540,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aab(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_540
tff(fact_8723_ATP_Olambda__541,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wc(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_541
tff(fact_8724_ATP_Olambda__542,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aac(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_542
tff(fact_8725_ATP_Olambda__543,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zd(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_543
tff(fact_8726_ATP_Olambda__544,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tz(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_544
tff(fact_8727_ATP_Olambda__545,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,product_prod(B,B),aa(A,fun(A,product_prod(B,B)),aTP_Lamp_ady(fun(A,B),fun(A,fun(A,product_prod(B,B))),Uu),Uua),Uub) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uu,Uua)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_545
tff(fact_8728_ATP_Olambda__546,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_ado(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).

% ATP.lambda_546
tff(fact_8729_ATP_Olambda__547,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_zx(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_547
tff(fact_8730_ATP_Olambda__548,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_zb(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_548
tff(fact_8731_ATP_Olambda__549,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ua(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_549
tff(fact_8732_ATP_Olambda__550,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_adh(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ).

% ATP.lambda_550
tff(fact_8733_ATP_Olambda__551,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,$o,Uu,Uub)
            | aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_551
tff(fact_8734_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aij(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_552
tff(fact_8735_ATP_Olambda__553,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aay(A,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_553
tff(fact_8736_ATP_Olambda__554,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ye(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_554
tff(fact_8737_ATP_Olambda__555,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_we(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_555
tff(fact_8738_ATP_Olambda__556,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_qe(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_556
tff(fact_8739_ATP_Olambda__557,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_ts(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_557
tff(fact_8740_ATP_Olambda__558,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ag(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_558
tff(fact_8741_ATP_Olambda__559,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_vt(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_559
tff(fact_8742_ATP_Olambda__560,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xp(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_560
tff(fact_8743_ATP_Olambda__561,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wa(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_561
tff(fact_8744_ATP_Olambda__562,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xm(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_562
tff(fact_8745_ATP_Olambda__563,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_bp(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_563
tff(fact_8746_ATP_Olambda__564,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(real,A),Uua: A,Uub: real] : aa(real,A,aa(A,fun(real,A),aTP_Lamp_agy(fun(real,A),fun(A,fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_564
tff(fact_8747_ATP_Olambda__565,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_565
tff(fact_8748_ATP_Olambda__566,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_bo(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_566
tff(fact_8749_ATP_Olambda__567,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_px(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_567
tff(fact_8750_ATP_Olambda__568,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yi(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_568
tff(fact_8751_ATP_Olambda__569,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_to(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_569
tff(fact_8752_ATP_Olambda__570,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_un(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_570
tff(fact_8753_ATP_Olambda__571,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_vx(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_571
tff(fact_8754_ATP_Olambda__572,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_vv(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_572
tff(fact_8755_ATP_Olambda__573,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xn(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_573
tff(fact_8756_ATP_Olambda__574,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(real,A),Uua: A,Uub: real] : aa(real,A,aa(A,fun(real,A),aTP_Lamp_agw(fun(real,A),fun(A,fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_574
tff(fact_8757_ATP_Olambda__575,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: 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)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_575
tff(fact_8758_ATP_Olambda__576,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_bk(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_576
tff(fact_8759_ATP_Olambda__577,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_rk(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_577
tff(fact_8760_ATP_Olambda__578,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_zh(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_578
tff(fact_8761_ATP_Olambda__579,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_zn(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_579
tff(fact_8762_ATP_Olambda__580,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pt(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_580
tff(fact_8763_ATP_Olambda__581,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ys(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_581
tff(fact_8764_ATP_Olambda__582,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_tl(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_582
tff(fact_8765_ATP_Olambda__583,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_tg(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_583
tff(fact_8766_ATP_Olambda__584,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ec(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_584
tff(fact_8767_ATP_Olambda__585,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_um(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_585
tff(fact_8768_ATP_Olambda__586,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ta(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_586
tff(fact_8769_ATP_Olambda__587,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_587
tff(fact_8770_ATP_Olambda__588,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_qi(fun(real,real),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uu,Uub)),Uua) ).

% ATP.lambda_588
tff(fact_8771_ATP_Olambda__589,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: nat,Uub: B] : aa(B,A,aa(nat,fun(B,A),aTP_Lamp_fa(fun(B,A),fun(nat,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_589
tff(fact_8772_ATP_Olambda__590,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_rt(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_590
tff(fact_8773_ATP_Olambda__591,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_zt(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_591
tff(fact_8774_ATP_Olambda__592,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_qf(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_592
tff(fact_8775_ATP_Olambda__593,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_yz(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_593
tff(fact_8776_ATP_Olambda__594,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_wz(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_594
tff(fact_8777_ATP_Olambda__595,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_ty(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_595
tff(fact_8778_ATP_Olambda__596,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_tu(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_596
tff(fact_8779_ATP_Olambda__597,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xk(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_597
tff(fact_8780_ATP_Olambda__598,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tb(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_598
tff(fact_8781_ATP_Olambda__599,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,A),Uua: A,Uub: real] : aa(real,A,aa(A,fun(real,A),aTP_Lamp_aha(fun(real,A),fun(A,fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_599
tff(fact_8782_ATP_Olambda__600,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_xt(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_600
tff(fact_8783_ATP_Olambda__601,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_afa(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_601
tff(fact_8784_ATP_Olambda__602,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ip(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_602
tff(fact_8785_ATP_Olambda__603,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_qm(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_603
tff(fact_8786_ATP_Olambda__604,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wl(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_604
tff(fact_8787_ATP_Olambda__605,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: int,Uub: B] : aa(B,A,aa(int,fun(B,A),aTP_Lamp_aaq(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_605
tff(fact_8788_ATP_Olambda__606,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_aap(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_606
tff(fact_8789_ATP_Olambda__607,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,option(B)),Uua: fun(B,option(A)),Uub: C] : aa(C,option(A),aa(fun(B,option(A)),fun(C,option(A)),aTP_Lamp_ahs(fun(C,option(B)),fun(fun(B,option(A)),fun(C,option(A))),Uu),Uua),Uub) = aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(C,option(B),Uu,Uub)),Uua) ).

% ATP.lambda_607
tff(fact_8790_ATP_Olambda__608,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ahd(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu) ) ) ).

% ATP.lambda_608
tff(fact_8791_ATP_Olambda__609,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aev(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != 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),Uua),Uu) ) ) ).

% ATP.lambda_609
tff(fact_8792_ATP_Olambda__610,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jj(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub))) ) ).

% ATP.lambda_610
tff(fact_8793_ATP_Olambda__611,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_du(real,fun(fun(nat,A),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_611
tff(fact_8794_ATP_Olambda__612,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yd(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_612
tff(fact_8795_ATP_Olambda__613,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_pd(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),Uu),Uua),Uub) = aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(rat,rat,inverse_inverse(rat),aa(nat,rat,Uu,Uub))),aa(rat,rat,inverse_inverse(rat),aa(nat,rat,Uua,Uub))) ).

% ATP.lambda_613
tff(fact_8796_ATP_Olambda__614,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_xr(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua) ) ) ).

% ATP.lambda_614
tff(fact_8797_ATP_Olambda__615,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_ahf(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua) ) ) ).

% ATP.lambda_615
tff(fact_8798_ATP_Olambda__616,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hz(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_616
tff(fact_8799_ATP_Olambda__617,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wd(fun(A,B),fun(B,fun(A,$o)),Uu),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,Uu,Uub)) ) ) ).

% ATP.lambda_617
tff(fact_8800_ATP_Olambda__618,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahk(fun(A,A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( ? [X: A] :
                ( ( Uub = aa(A,A,Uu,X) )
                & aa(A,$o,Uua,X) )
            | ? [M8: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X: A] :
                    ( member(A,X,M8)
                   => aa(A,$o,Uua,X) ) ) ) ) ) ).

% ATP.lambda_618
tff(fact_8801_ATP_Olambda__619,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),$o)),Uua: list(A),Uub: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_aad(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub)
        <=> ( ? [Y3: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) ) )
            | ? [X: A,Y3: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) )
            | ? [X: A,Y3: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uu,Xs3),Ys4) ) ) ) ) ).

% ATP.lambda_619
tff(fact_8802_ATP_Olambda__620,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_acn(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( finite_finite(A,Uua)
        & finite_finite(A,Uub)
        & ( Uub != bot_bot(set(A)) )
        & ! [X: A] :
            ( member(A,X,Uua)
           => ? [Xa2: A] :
                ( member(A,Xa2,Uub)
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa2),Uu) ) ) ) ) ).

% ATP.lambda_620
tff(fact_8803_ATP_Olambda__621,axiom,
    ! [A: $tType,Uu: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),Uua: fun($o,fun($o,A)),Uub: vEBT_VEBT] : aa(vEBT_VEBT,product_prod(vEBT_VEBT,A),aa(fun($o,fun($o,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_afd(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),Uu),Uua),Uub) = aa(A,product_prod(vEBT_VEBT,A),aa(vEBT_VEBT,fun(A,product_prod(vEBT_VEBT,A)),product_Pair(vEBT_VEBT,A),Uub),vEBT_rec_VEBT(A,Uu,Uua,Uub)) ).

% ATP.lambda_621
tff(fact_8804_ATP_Olambda__622,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_622
tff(fact_8805_ATP_Olambda__623,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jg(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_623
tff(fact_8806_ATP_Olambda__624,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_je(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_624
tff(fact_8807_ATP_Olambda__625,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_aal(A,fun(int,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uua)),power_int(A,Uu,aa(int,int,aa(int,fun(int,int),minus_minus(int),Uua),one_one(int))))) ) ).

% ATP.lambda_625
tff(fact_8808_ATP_Olambda__626,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ci(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_626
tff(fact_8809_ATP_Olambda__627,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ch(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_627
tff(fact_8810_ATP_Olambda__628,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cw(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_628
tff(fact_8811_ATP_Olambda__629,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_uy(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Uua),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_629
tff(fact_8812_ATP_Olambda__630,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bv(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_630
tff(fact_8813_ATP_Olambda__631,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_cf(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_631
tff(fact_8814_ATP_Olambda__632,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: 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_nk(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu),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)),Uu),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).

% ATP.lambda_632
tff(fact_8815_ATP_Olambda__633,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: 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_ni(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu),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),Uu),Uub)) ).

% ATP.lambda_633
tff(fact_8816_ATP_Olambda__634,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_wi(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> member(real,Uub,set_or5935395276787703475ssThan(real,Uu,Uua)) ) ).

% ATP.lambda_634
tff(fact_8817_ATP_Olambda__635,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_acx(fun(A,B),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
        <=> member(B,Uub,image(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_635
tff(fact_8818_ATP_Olambda__636,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_vz(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_636
tff(fact_8819_ATP_Olambda__637,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_vs(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_637
tff(fact_8820_ATP_Olambda__638,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wh(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_638
tff(fact_8821_ATP_Olambda__639,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_639
tff(fact_8822_ATP_Olambda__640,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aau(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_640
tff(fact_8823_ATP_Olambda__641,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_vw(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_641
tff(fact_8824_ATP_Olambda__642,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_om(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_642
tff(fact_8825_ATP_Olambda__643,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wt(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_643
tff(fact_8826_ATP_Olambda__644,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_vu(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_644
tff(fact_8827_ATP_Olambda__645,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zi(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_645
tff(fact_8828_ATP_Olambda__646,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_de(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_646
tff(fact_8829_ATP_Olambda__647,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bj(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_647
tff(fact_8830_ATP_Olambda__648,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ed(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_648
tff(fact_8831_ATP_Olambda__649,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ul(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_649
tff(fact_8832_ATP_Olambda__650,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sz(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_650
tff(fact_8833_ATP_Olambda__651,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(real,A),Uua: A,Uub: real] : aa(real,A,aa(A,fun(real,A),aTP_Lamp_agx(fun(real,A),fun(A,fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(real,A,Uu,Uub)) ) ).

% ATP.lambda_651
tff(fact_8834_ATP_Olambda__652,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_652
tff(fact_8835_ATP_Olambda__653,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_rl(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_653
tff(fact_8836_ATP_Olambda__654,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_zo(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_654
tff(fact_8837_ATP_Olambda__655,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ps(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_655
tff(fact_8838_ATP_Olambda__656,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yt(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_656
tff(fact_8839_ATP_Olambda__657,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_tm(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_657
tff(fact_8840_ATP_Olambda__658,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_th(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_658
tff(fact_8841_ATP_Olambda__659,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xv(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_659
tff(fact_8842_ATP_Olambda__660,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_fe(A,fun(fun(B,nat),fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_660
tff(fact_8843_ATP_Olambda__661,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vc(fun(A,nat),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_661
tff(fact_8844_ATP_Olambda__662,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tt(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_662
tff(fact_8845_ATP_Olambda__663,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afb(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_663
tff(fact_8846_ATP_Olambda__664,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yj(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> dvd_dvd(B,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_664
tff(fact_8847_ATP_Olambda__665,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_mz(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu,Uub)) ).

% ATP.lambda_665
tff(fact_8848_ATP_Olambda__666,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_zw(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_666
tff(fact_8849_ATP_Olambda__667,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_za(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_667
tff(fact_8850_ATP_Olambda__668,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_tx(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ).

% ATP.lambda_668
tff(fact_8851_ATP_Olambda__669,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_nv(fun(list(A),A),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_669
tff(fact_8852_ATP_Olambda__670,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fl(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_670
tff(fact_8853_ATP_Olambda__671,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_xa(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_671
tff(fact_8854_ATP_Olambda__672,axiom,
    ! [Uu: real,Uua: real,Uub: product_unit] : aa(product_unit,real,aa(real,fun(product_unit,real),aTP_Lamp_agb(real,fun(real,fun(product_unit,real)),Uu),Uua),Uub) = powr_real(Uu,Uua) ).

% ATP.lambda_672
tff(fact_8855_ATP_Olambda__673,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_aea(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert(B,Uu),Uua) ).

% ATP.lambda_673
tff(fact_8856_ATP_Olambda__674,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_aef(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = image(D,B,Uu,Uua) ).

% ATP.lambda_674
tff(fact_8857_ATP_Olambda__675,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_us(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_675
tff(fact_8858_ATP_Olambda__676,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_td(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_676
tff(fact_8859_ATP_Olambda__677,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qa(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_677
tff(fact_8860_ATP_Olambda__678,axiom,
    ! [Uu: fun(nat,real),Uua: nat,Uub: nat] : aa(nat,real,aa(nat,fun(nat,real),aTP_Lamp_aey(fun(nat,real),fun(nat,fun(nat,real)),Uu),Uua),Uub) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_678
tff(fact_8861_ATP_Olambda__679,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_vn(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_679
tff(fact_8862_ATP_Olambda__680,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_pe(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu),Uua),Uub) = aa(nat,set(A),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_680
tff(fact_8863_ATP_Olambda__681,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_681
tff(fact_8864_ATP_Olambda__682,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_uo(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_682
tff(fact_8865_ATP_Olambda__683,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xu(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_683
tff(fact_8866_ATP_Olambda__684,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ff(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_684
tff(fact_8867_ATP_Olambda__685,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_685
tff(fact_8868_ATP_Olambda__686,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_vy(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_686
tff(fact_8869_ATP_Olambda__687,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_tc(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_687
tff(fact_8870_ATP_Olambda__688,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_uh(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_688
tff(fact_8871_ATP_Olambda__689,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pm(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_689
tff(fact_8872_ATP_Olambda__690,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_ax(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_690
tff(fact_8873_ATP_Olambda__691,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_add(fun(product_prod(A,A),$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(product_prod(A,A),$o,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)) ) ) ).

% ATP.lambda_691
tff(fact_8874_ATP_Olambda__692,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ub(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),Uub)) ) ).

% ATP.lambda_692
tff(fact_8875_ATP_Olambda__693,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ze(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_693
tff(fact_8876_ATP_Olambda__694,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eg(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu)) ) ).

% ATP.lambda_694
tff(fact_8877_ATP_Olambda__695,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_aby(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_695
tff(fact_8878_ATP_Olambda__696,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,B),Uua: fun(A,C),Uub: A] : aa(A,B,aa(fun(A,C),fun(A,B),aTP_Lamp_abu(fun(C,B),fun(fun(A,C),fun(A,B)),Uu),Uua),Uub) = aa(C,B,Uu,aa(A,C,Uua,Uub)) ).

% ATP.lambda_696
tff(fact_8879_ATP_Olambda__697,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(num,B),Uub: num] : aa(num,A,aa(fun(num,B),fun(num,A),aTP_Lamp_me(fun(B,A),fun(fun(num,B),fun(num,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(num,B,Uua,Uub)) ).

% ATP.lambda_697
tff(fact_8880_ATP_Olambda__698,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_km(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_698
tff(fact_8881_ATP_Olambda__699,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_lr(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_699
tff(fact_8882_ATP_Olambda__700,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afv(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_700
tff(fact_8883_ATP_Olambda__701,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_xw(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_701
tff(fact_8884_ATP_Olambda__702,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_xx(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_702
tff(fact_8885_ATP_Olambda__703,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_po(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_703
tff(fact_8886_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_afs(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_704
tff(fact_8887_ATP_Olambda__705,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_zk(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_705
tff(fact_8888_ATP_Olambda__706,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,$o),aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_aib(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),Uu),Uua),Uub) = aa(A,fun(B,$o),Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_706
tff(fact_8889_ATP_Olambda__707,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ue(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_707
tff(fact_8890_ATP_Olambda__708,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pp(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_708
tff(fact_8891_ATP_Olambda__709,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xh(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_709
tff(fact_8892_ATP_Olambda__710,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ix(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_710
tff(fact_8893_ATP_Olambda__711,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_kc(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_711
tff(fact_8894_ATP_Olambda__712,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),$o),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),$o),aa(A,fun(A,fun(product_prod(A,A),$o)),aTP_Lamp_adf(fun(product_prod(A,A),$o),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_ade(fun(product_prod(A,A),$o),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ) ).

% ATP.lambda_712
tff(fact_8895_ATP_Olambda__713,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),$o),aa(A,fun(A,fun(product_prod(A,A),$o)),aTP_Lamp_acm(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_acl(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ).

% ATP.lambda_713
tff(fact_8896_ATP_Olambda__714,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_yr(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_yq(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub)) ) ).

% ATP.lambda_714
tff(fact_8897_ATP_Olambda__715,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_sy(fun(nat,A),fun(A,fun(A,A)),Uu),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_sx(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_715
tff(fact_8898_ATP_Olambda__716,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: A] : aa(A,option(A),aa(A,fun(A,option(A)),aTP_Lamp_aht(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Uu,Uua),Uub)) ).

% ATP.lambda_716
tff(fact_8899_ATP_Olambda__717,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_dk(fun(nat,A),fun(A,fun(nat,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub))) ) ).

% ATP.lambda_717
tff(fact_8900_ATP_Olambda__718,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_vg(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_718
tff(fact_8901_ATP_Olambda__719,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yk(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ~ dvd_dvd(B,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_719
tff(fact_8902_ATP_Olambda__720,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_ka(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_jz(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_720
tff(fact_8903_ATP_Olambda__721,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_mn(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [I4: 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,Uu),I4)),aa(nat,B,nth(B,Uua),I4)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).

% ATP.lambda_721
tff(fact_8904_ATP_Olambda__722,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_nf(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [I4: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu))
          & member(nat,I4,Uua) ) ) ).

% ATP.lambda_722
tff(fact_8905_ATP_Olambda__723,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_acs(fun(B,option(A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( member(B,X,Uua)
          & ( aa(B,option(A),Uu,X) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_723
tff(fact_8906_ATP_Olambda__724,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_vm(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),N4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,Uub,N4)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_724
tff(fact_8907_ATP_Olambda__725,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_wv(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [A6: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),A6)
             => ! [B5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A6),B5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or3652927894154168847AtMost(nat,A6,B5)))),aa(nat,real,Uua,A6)) ) ) ) ) ).

% ATP.lambda_725
tff(fact_8908_ATP_Olambda__726,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aah(fun(A,A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ? [N4: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N4),Uu),Uua) ) ).

% ATP.lambda_726
tff(fact_8909_ATP_Olambda__727,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_agd(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
    <=> ? [A16: B,A25: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A16)),aa(B,A,Uua,A25)) )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A16),A25),Uu) ) ) ).

% ATP.lambda_727
tff(fact_8910_ATP_Olambda__728,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_sr(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [A6: A,V5: list(A)] :
          ( ( Uub = append(A,Uua,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),V5)) )
          | ? [U4: list(A),Aa4: A,B5: A,Va4: list(A),W4: 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),Aa4),B5),Uu)
              & ( Uua = append(A,U4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa4),Va4)) )
              & ( Uub = append(A,U4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B5),W4)) ) ) ) ) ).

% ATP.lambda_728
tff(fact_8911_ATP_Olambda__729,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aho(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [X: A,Y3: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3) )
              & member(A,X,real_Vector_span(A,Uu))
              & member(A,Y3,real_Vector_span(A,Uua)) ) ) ) ).

% ATP.lambda_729
tff(fact_8912_ATP_Olambda__730,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_sq(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [Us3: list(A),Z2: A,Z8: A,Vs3: list(A)] :
          ( ( Uua = append(A,Us3,aa(list(A),list(A),aa(A,fun(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),Z8),Uu)
          & ( Uub = append(A,Us3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z8),Vs3)) ) ) ) ).

% ATP.lambda_730
tff(fact_8913_ATP_Olambda__731,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ht(A,fun(A,fun(nat,fun(nat,A))),Uu),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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_731
tff(fact_8914_ATP_Olambda__732,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hp(A,fun(A,fun(nat,fun(nat,A))),Uu),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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_732
tff(fact_8915_ATP_Olambda__733,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hr(A,fun(A,fun(nat,fun(nat,A))),Uu),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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_733
tff(fact_8916_ATP_Olambda__734,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] :
      aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_abb(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = $ite(aa(product_prod(C,A),A,product_snd(C,A),Uu) = Uua,aa(set(product_prod(C,B)),set(product_prod(C,B)),insert(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_734
tff(fact_8917_ATP_Olambda__735,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_735
tff(fact_8918_ATP_Olambda__736,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_he(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_736
tff(fact_8919_ATP_Olambda__737,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_zg(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_737
tff(fact_8920_ATP_Olambda__738,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hf(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_738
tff(fact_8921_ATP_Olambda__739,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_739
tff(fact_8922_ATP_Olambda__740,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: set(nat),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ej(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(member(nat,Uuc,Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_740
tff(fact_8923_ATP_Olambda__741,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_fj(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_741
tff(fact_8924_ATP_Olambda__742,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_cg(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_742
tff(fact_8925_ATP_Olambda__743,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_kf(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_743
tff(fact_8926_ATP_Olambda__744,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_yh(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_744
tff(fact_8927_ATP_Olambda__745,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_yg(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_745
tff(fact_8928_ATP_Olambda__746,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_abh(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_abg(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_746
tff(fact_8929_ATP_Olambda__747,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_aba(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_aaz(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_747
tff(fact_8930_ATP_Olambda__748,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_la(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uuc),aa(A,fun(A,A),times_times(A),Uua)),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))) ) ).

% ATP.lambda_748
tff(fact_8931_ATP_Olambda__749,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(real,A),Uub: fun(real,B),Uuc: real] : aa(real,C,aa(fun(real,B),fun(real,C),aa(fun(real,A),fun(fun(real,B),fun(real,C)),aTP_Lamp_ahc(fun(A,fun(B,C)),fun(fun(real,A),fun(fun(real,B),fun(real,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(real,A,Uua,Uuc)),aa(real,B,Uub,Uuc)) ) ).

% ATP.lambda_749
tff(fact_8932_ATP_Olambda__750,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: fun(D,B),Uuc: D] : aa(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_aft(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),aa(D,B,Uub,Uuc)) ) ).

% ATP.lambda_750
tff(fact_8933_ATP_Olambda__751,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,B),Uub: A,Uuc: C] :
      ( aa(C,$o,aa(A,fun(C,$o),aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aia(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_751
tff(fact_8934_ATP_Olambda__752,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cm(A,fun(A,fun(nat,fun(nat,A))),Uu),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_752
tff(fact_8935_ATP_Olambda__753,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_bm(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_bl(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_753
tff(fact_8936_ATP_Olambda__754,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: nat] :
          aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_hi(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(
                Uuc = zero_zero(nat),
                aa(A,A,uminus_uminus(A),Uub),
                $ite(Uuc = Uu,one_one(A),zero_zero(A)) )),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_754
tff(fact_8937_ATP_Olambda__755,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,fun(A,$o)),Uub: set(A),Uuc: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_ahn(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,pred_chain(A,Uu,Uua),Uuc)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uub),Uuc) ) ) ).

% ATP.lambda_755
tff(fact_8938_ATP_Olambda__756,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_hb(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_ha(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_756
tff(fact_8939_ATP_Olambda__757,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_gx(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_gw(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_757
tff(fact_8940_ATP_Olambda__758,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_qy(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_758
tff(fact_8941_ATP_Olambda__759,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_qw(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_759
tff(fact_8942_ATP_Olambda__760,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qu(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uua)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uub),Uua)),Uuc)) ).

% ATP.lambda_760
tff(fact_8943_ATP_Olambda__761,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qv(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uub)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uua),Uub)),Uuc)) ).

% ATP.lambda_761
tff(fact_8944_ATP_Olambda__762,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cs(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_762
tff(fact_8945_ATP_Olambda__763,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: set(A),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_aet(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),product_Sigma(A,A,Uua,aTP_Lamp_aec(set(A),fun(A,set(A)),Uua)))
        & aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uuc) ) ) ).

% ATP.lambda_763
tff(fact_8946_ATP_Olambda__764,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lc(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).

% ATP.lambda_764
tff(fact_8947_ATP_Olambda__765,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cm(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Uuc))) ) ).

% ATP.lambda_765
tff(fact_8948_ATP_Olambda__766,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_vd(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu,Uua))),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_766
tff(fact_8949_ATP_Olambda__767,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_sn(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
          & 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_767
tff(fact_8950_ATP_Olambda__768,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_agu(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc)),Uu) ) ).

% ATP.lambda_768
tff(fact_8951_ATP_Olambda__769,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_xq(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_769
tff(fact_8952_ATP_Olambda__770,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_xg(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_770
tff(fact_8953_ATP_Olambda__771,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gb(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_771
tff(fact_8954_ATP_Olambda__772,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fz(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_772
tff(fact_8955_ATP_Olambda__773,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ga(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_773
tff(fact_8956_ATP_Olambda__774,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hx(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_774
tff(fact_8957_ATP_Olambda__775,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_st(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X: A,Y3: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uub = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs5)) )
            & ( Uuc = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),Uu) ) ) ) ).

% ATP.lambda_775
tff(fact_8958_ATP_Olambda__776,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_cu(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)) ) ).

% ATP.lambda_776
tff(fact_8959_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_jz(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_777
tff(fact_8960_ATP_Olambda__778,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_jy(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_778
tff(fact_8961_ATP_Olambda__779,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ct(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_779
tff(fact_8962_ATP_Olambda__780,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: fun(A,real),Uuc: A] :
          ( aa(A,$o,aa(fun(A,real),fun(A,$o),aa(fun(A,real),fun(fun(A,real),fun(A,$o)),aTP_Lamp_zz(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uua,Uuc)),aa(A,real,Uub,Uuc)) ) ) ) ).

% ATP.lambda_780
tff(fact_8963_ATP_Olambda__781,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_kd(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_781
tff(fact_8964_ATP_Olambda__782,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_cv(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uuc))) ) ).

% ATP.lambda_782
tff(fact_8965_ATP_Olambda__783,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fy(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_783
tff(fact_8966_ATP_Olambda__784,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_lh(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_784
tff(fact_8967_ATP_Olambda__785,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_ky(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_785
tff(fact_8968_ATP_Olambda__786,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_lf(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_786
tff(fact_8969_ATP_Olambda__787,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lj(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ).

% ATP.lambda_787
tff(fact_8970_ATP_Olambda__788,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_ll(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).

% ATP.lambda_788
tff(fact_8971_ATP_Olambda__789,axiom,
    ! [A: $tType,B: $tType,Uu: 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_yn(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),Uu),Uub)),zip(A,B,Uua,Uuc)) ).

% ATP.lambda_789
tff(fact_8972_ATP_Olambda__790,axiom,
    ! [A: $tType,B: $tType,Uu: 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_yo(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),Uu)),zip(A,B,Uuc,Uua)) ).

% ATP.lambda_790
tff(fact_8973_ATP_Olambda__791,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_kt(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_791
tff(fact_8974_ATP_Olambda__792,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_acq(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua) ) ) ).

% ATP.lambda_792
tff(fact_8975_ATP_Olambda__793,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_oz(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu),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)),Uu,Uua)) ) ).

% ATP.lambda_793
tff(fact_8976_ATP_Olambda__794,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ar(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_794
tff(fact_8977_ATP_Olambda__795,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_at(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_795
tff(fact_8978_ATP_Olambda__796,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_vl(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))) ) ).

% ATP.lambda_796
tff(fact_8979_ATP_Olambda__797,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dq(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_797
tff(fact_8980_ATP_Olambda__798,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_jm(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_798
tff(fact_8981_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_sx(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc))),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_799
tff(fact_8982_ATP_Olambda__800,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A,Uuc: nat] : aa(nat,B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_yq(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uua,Uuc)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_800
tff(fact_8983_ATP_Olambda__801,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu,Uua)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_801
tff(fact_8984_ATP_Olambda__802,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hn(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_802
tff(fact_8985_ATP_Olambda__803,axiom,
    ! [A: $tType,Uu: fun(A,$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,$o)),aTP_Lamp_acj(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uuc)
        & ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).

% ATP.lambda_803
tff(fact_8986_ATP_Olambda__804,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: A] : aa(A,A,aa(B,fun(A,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_nn(fun(B,A),fun(A,fun(B,fun(A,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).

% ATP.lambda_804
tff(fact_8987_ATP_Olambda__805,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_ha(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uuc)),aa(nat,nat,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_805
tff(fact_8988_ATP_Olambda__806,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_gu(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_806
tff(fact_8989_ATP_Olambda__807,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_gw(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_807
tff(fact_8990_ATP_Olambda__808,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_bl(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_808
tff(fact_8991_ATP_Olambda__809,axiom,
    ! [A: $tType,B: $tType,Uu: 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_aej(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu,Uub)),aa(B,nat,Uua,Uuc)) ).

% ATP.lambda_809
tff(fact_8992_ATP_Olambda__810,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_0(C)
     => ! [Uu: 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_aeq(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu,Uub)),aa(B,C,Uua,Uuc)) ) ).

% ATP.lambda_810
tff(fact_8993_ATP_Olambda__811,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_mx(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).

% ATP.lambda_811
tff(fact_8994_ATP_Olambda__812,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,D),Uub: A,Uuc: B] : aa(B,product_prod(C,D),aa(A,fun(B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_adc(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),Uu),Uua),Uub),Uuc) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,Uu,Uub)),aa(B,D,Uua,Uuc)) ).

% ATP.lambda_812
tff(fact_8995_ATP_Olambda__813,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_nu(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_813
tff(fact_8996_ATP_Olambda__814,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_814
tff(fact_8997_ATP_Olambda__815,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_si(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uub)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% ATP.lambda_815
tff(fact_8998_ATP_Olambda__816,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_rw(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_816
tff(fact_8999_ATP_Olambda__817,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_rg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% ATP.lambda_817
tff(fact_9000_ATP_Olambda__818,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sm(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ) ).

% ATP.lambda_818
tff(fact_9001_ATP_Olambda__819,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_wg(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu,Uub))),aa(A,B,Uua,Uuc)))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_819
tff(fact_9002_ATP_Olambda__820,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_ke(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_kd(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_820
tff(fact_9003_ATP_Olambda__821,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: 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_aep(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),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,Uu,aa(set(A),set(A),insert(A,Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_821
tff(fact_9004_ATP_Olambda__822,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_wu(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_dj(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),set_ord_atMost(nat,Uua)))) ) ) ).

% ATP.lambda_822
tff(fact_9005_ATP_Olambda__823,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_823
tff(fact_9006_ATP_Olambda__824,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jc(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_824
tff(fact_9007_ATP_Olambda__825,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_825
tff(fact_9008_ATP_Olambda__826,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_bw(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_826
tff(fact_9009_ATP_Olambda__827,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,Uu: 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_my(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,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_827
tff(fact_9010_ATP_Olambda__828,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(product_prod(B,B),$o),Uuc: A] :
          ( aa(A,$o,aa(fun(product_prod(B,B),$o),fun(A,$o),aa(B,fun(fun(product_prod(B,B),$o),fun(A,$o)),aTP_Lamp_adb(fun(A,B),fun(B,fun(fun(product_prod(B,B),$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(product_prod(B,B),$o,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uu,Uuc)),Uua)) ) ) ).

% ATP.lambda_828
tff(fact_9011_ATP_Olambda__829,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Uu: 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_mw(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,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_829
tff(fact_9012_ATP_Olambda__830,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_yc(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_yb(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_830
tff(fact_9013_ATP_Olambda__831,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_ya(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_xz(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_831
tff(fact_9014_ATP_Olambda__832,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_rs(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)))) ) ).

% ATP.lambda_832
tff(fact_9015_ATP_Olambda__833,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_sw(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [Y3: 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),Y3),Uu)
          & member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Uuc),Uua) ) ) ).

% ATP.lambda_833
tff(fact_9016_ATP_Olambda__834,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_mv(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),Uu),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,Uu) ) ) ).

% ATP.lambda_834
tff(fact_9017_ATP_Olambda__835,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aa(fun(A,B),fun(product_prod(A,A),$o),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o)),aTP_Lamp_ahe(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),Uu),Uua),Uub),Uuc)
    <=> ? [A16: A,A25: A] :
          ( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A16),A25) )
          & member(A,A16,Uu)
          & member(A,A25,Uu)
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uub,A16)),aa(A,B,Uub,A25)),Uua) ) ) ).

% ATP.lambda_835
tff(fact_9018_ATP_Olambda__836,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] :
      aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_abg(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(C,B)),set(product_prod(C,B)),insert(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_836
tff(fact_9019_ATP_Olambda__837,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] :
      aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_aaz(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,C)),set(product_prod(A,C)),insert(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_837
tff(fact_9020_ATP_Olambda__838,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B] : aa(B,C,aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_sc(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_sb(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_838
tff(fact_9021_ATP_Olambda__839,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_hm(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_hl(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),Uu),Uud))) ) ).

% ATP.lambda_839
tff(fact_9022_ATP_Olambda__840,axiom,
    ! [Uu: nat,Uua: fun(nat,fun(real,real)),Uub: real,Uuc: nat,Uud: real] : aa(real,real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_qx(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_qw(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),Uu),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uud),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)))))) ).

% ATP.lambda_840
tff(fact_9023_ATP_Olambda__841,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_ho(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hn(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud)) ) ).

% ATP.lambda_841
tff(fact_9024_ATP_Olambda__842,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hk(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_842
tff(fact_9025_ATP_Olambda__843,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hg(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_843
tff(fact_9026_ATP_Olambda__844,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abo(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),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),Uu),transitive_trancl(A,Uub))
          | ( Uuc = Uu ) )
        & ( 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_844
tff(fact_9027_ATP_Olambda__845,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hh(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_845
tff(fact_9028_ATP_Olambda__846,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_acl(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Uua),aa(set(A),set(A),insert(A,Uub),aa(set(A),set(A),insert(A,Uuc),aa(set(A),set(A),insert(A,Uud),bot_bot(set(A))))))),field2(A,Uu))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud)),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))),Uu),id2(A)))
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,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),Uua),Uuc),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))),Uu),id2(A))) )
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & ( Uua = 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),Uud),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))),Uu),id2(A))) ) ) ) ) ).

% ATP.lambda_846
tff(fact_9029_ATP_Olambda__847,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abx(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),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),Uu),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_847
tff(fact_9030_ATP_Olambda__848,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_hl(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uud)),one_one(nat)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_848
tff(fact_9031_ATP_Olambda__849,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: nat,Uud: A] : aa(A,B,aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_ru(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uuc)),aa(A,B,Uua,Uud))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),one_one(nat)))) ) ).

% ATP.lambda_849
tff(fact_9032_ATP_Olambda__850,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(product_prod(B,B),$o),Uuc: A,Uud: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aa(fun(product_prod(B,B),$o),fun(A,fun(A,$o)),aa(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o))),aTP_Lamp_adg(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ( member(A,Uuc,Uu)
           => ( member(A,Uud,Uu)
             => aa(product_prod(B,B),$o,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uuc)),aa(A,B,Uua,Uud))) ) ) ) ) ).

% ATP.lambda_850
tff(fact_9033_ATP_Olambda__851,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),$o),Uua: A,Uub: A,Uuc: A,Uud: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_ade(fun(product_prod(A,A),$o),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ( ( Uub = Uuc )
           => aa(product_prod(A,A),$o,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)) ) ) ) ).

% ATP.lambda_851
tff(fact_9034_ATP_Olambda__852,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: int,Uud: B] : aa(B,A,aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_aar(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uub,Uud)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uuc)),power_int(A,aa(B,A,Uu,Uua),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uuc),one_one(int))))) ) ).

% ATP.lambda_852
tff(fact_9035_ATP_Olambda__853,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_0(C)
     => ! [Uu: fun(A,C),Uua: fun(B,C),Uub: set(A),Uuc: set(B),Uud: C] :
          ( aa(C,$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,$o))),aTP_Lamp_aer(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ? [A6: A,B5: B] :
              ( ( Uud = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu,A6)),aa(B,C,Uua,B5)) )
              & member(A,A6,Uub)
              & member(B,B5,Uuc) ) ) ) ).

% ATP.lambda_853
tff(fact_9036_ATP_Olambda__854,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B,Uue: A] : aa(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_sb(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,aa(A,fun(B,C),Uub,Uue),Uud)),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_rz(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(set(A),set(A),insert(A,Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_854
tff(fact_9037_ATP_Olambda__855,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_rp(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uuc,Uub)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_855
tff(fact_9038_ATP_Olambda__856,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_xz(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),Uu)
        | ( ( Uub = Uud )
          & 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_856
tff(fact_9039_ATP_Olambda__857,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_se(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu,Uub),aa(A,real,Uuc,Uub))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uud,Uue)),aa(real,real,ln_ln(real),aa(A,real,Uu,Uub)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub))),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_857
tff(fact_9040_ATP_Olambda__858,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_ri(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_858
tff(fact_9041_ATP_Olambda__859,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_rn(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_859
tff(fact_9042_ATP_Olambda__860,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_ry(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))),aa(A,B,Uud,Uue))),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_860
tff(fact_9043_ATP_Olambda__861,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_yb(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa(A,$o,Uu,Uud)
        & 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_861
tff(fact_9044_ATP_Olambda__862,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: fun(D,A),Uuc: D,Uud: fun(D,B),Uue: fun(D,B),Uuf: D] : aa(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,B),fun(fun(D,B),fun(D,C)),aa(D,fun(fun(D,B),fun(fun(D,B),fun(D,C))),aa(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C)))),aa(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C))))),aTP_Lamp_afu(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),aa(D,B,Uue,Uuf))),aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uub,Uuf)),aa(D,B,Uud,Uuc))) ) ).

% ATP.lambda_862
tff(fact_9045_ATP_Olambda__863,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ms(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uue),Uuf)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuf),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uud))
           => ( ? [X_1: nat] : vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I4),X_1)
            <=> vEBT_V8194947554948674370ptions(Uub,I4) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X7: nat] : vEBT_V8194947554948674370ptions(X,X7) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),X)
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_863
tff(fact_9046_ATP_Olambda__864,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_aee(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_864
tff(fact_9047_ATP_Olambda__865,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_aeb(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_865
tff(fact_9048_ATP_Olambda__866,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_adz(set(A),fun(A,set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_866
tff(fact_9049_ATP_Olambda__867,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_aei(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_867
tff(fact_9050_ATP_Olambda__868,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_aed(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_868
tff(fact_9051_ATP_Olambda__869,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_aec(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_869
tff(fact_9052_ATP_Olambda__870,axiom,
    ! [Uu: rat,Uua: nat] : aa(nat,rat,aTP_Lamp_ot(rat,fun(nat,rat),Uu),Uua) = Uu ).

% ATP.lambda_870
tff(fact_9053_ATP_Olambda__871,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_yl(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_871
tff(fact_9054_ATP_Olambda__872,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_js(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_872
tff(fact_9055_ATP_Olambda__873,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_jt(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_873
tff(fact_9056_ATP_Olambda__874,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_nb(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_874
tff(fact_9057_ATP_Olambda__875,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_bx(complex,complex),Uu) = Uu ).

% ATP.lambda_875
tff(fact_9058_ATP_Olambda__876,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ce(nat,nat),Uu) = Uu ).

% ATP.lambda_876
tff(fact_9059_ATP_Olambda__877,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_bh(int,int),Uu) = Uu ).

% ATP.lambda_877
tff(fact_9060_ATP_Olambda__878,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_py(A,A),Uu) = Uu ) ).

% ATP.lambda_878
tff(fact_9061_ATP_Olambda__879,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aae(A,A),Uu) = Uu ) ).

% ATP.lambda_879
tff(fact_9062_ATP_Olambda__880,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ov(A,A),Uu) = Uu ) ).

% ATP.lambda_880
tff(fact_9063_ATP_Olambda__881,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ab(A,A),Uu) = Uu ) ).

% ATP.lambda_881
tff(fact_9064_ATP_Olambda__882,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_mf(A,A),Uu) = Uu ).

% ATP.lambda_882
tff(fact_9065_ATP_Olambda__883,axiom,
    ! [Uu: nat] : aa(nat,rat,aTP_Lamp_pa(nat,rat),Uu) = zero_zero(rat) ).

% ATP.lambda_883
tff(fact_9066_ATP_Olambda__884,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ae(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_884
tff(fact_9067_ATP_Olambda__885,axiom,
    ! [Uu: nat] : aa(nat,rat,aTP_Lamp_pb(nat,rat),Uu) = one_one(rat) ).

% ATP.lambda_885
tff(fact_9068_ATP_Olambda__886,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_ev(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_886
tff(fact_9069_ATP_Olambda__887,axiom,
    ! [A: $tType,Uu: A] : aa(A,real,aTP_Lamp_kb(A,real),Uu) = one_one(real) ).

% ATP.lambda_887
tff(fact_9070_ATP_Olambda__888,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_ju(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_888
tff(fact_9071_ATP_Olambda__889,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,option(B),aTP_Lamp_agc(C,option(B)),Uu) = none(B) ).

% ATP.lambda_889
tff(fact_9072_ATP_Olambda__890,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_nt(B,option(A)),Uu) = none(A) ).

% ATP.lambda_890
tff(fact_9073_ATP_Olambda__891,axiom,
    ! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_aas(A,option(C)),Uu) = none(C) ).

% ATP.lambda_891
tff(fact_9074_ATP_Olambda__892,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_on(A,option(B)),Uu) = none(B) ).

% ATP.lambda_892
tff(fact_9075_ATP_Olambda__893,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_ko(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_893
tff(fact_9076_ATP_Olambda__894,axiom,
    ! [B: $tType,Uu: B] :
      ( aa(B,$o,aTP_Lamp_aic(B,$o),Uu)
    <=> $false ) ).

% ATP.lambda_894
tff(fact_9077_ATP_Olambda__895,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_lw(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_895
tff(fact_9078_ATP_Olambda__896,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_kn(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_896
tff(fact_9079_ATP_Olambda__897,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_lv(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_897

% Type constructors (780)
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,
    ! [A17: $tType] : bounded_lattice(filter(A17)) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_3,axiom,
    bounded_lattice($o) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_4,axiom,
    ! [A17: $tType] : bounded_lattice(set(A17)) ).

tff(tcon_fun___Lattices_Obounded__lattice_5,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => condit1219197933456340205attice(fun(A17,A18)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( counta3822494911875563373attice(A18)
     => counta3822494911875563373attice(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple592849572758109894attice(A18)
     => comple592849572758109894attice(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4967611905675639751up_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4346867609351753570nf_top(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple6319245703460814977attice(fun(A17,A18)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A17: $tType,A18: $tType] :
      ( boolea8198339166811842893lgebra(A18)
     => boolea8198339166811842893lgebra(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple9053668089753744459l_ccpo(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A17: $tType,A18: $tType] :
      ( semilattice_sup(A18)
     => semilattice_sup(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A17: $tType,A18: $tType] :
      ( semilattice_inf(A18)
     => semilattice_inf(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order_top(A18)
     => order_top(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order_bot(A18)
     => order_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A17: $tType,A18: $tType] :
      ( preorder(A18)
     => preorder(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( lattice(A18)
     => lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order(A18)
     => order(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ord(A18)
     => ord(fun(A17,A18)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A17: $tType,A18: $tType] :
      ( uminus(A18)
     => uminus(fun(A17,A18)) ) ).

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_Onormalization__semidom,axiom,
    normal8620421768224518004emidom(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___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_9,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_10,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___Rings_Omult__zero,axiom,
    mult_zero(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring(int) ).

tff(tcon_Int_Oint___Orderings_Oorder_11,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_12,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_13,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___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_14,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_15,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_16,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_17,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_18,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_19,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_20,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_21,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_22,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_23,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_24,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_25,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_26,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_27,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_28,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_29,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_30,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_31,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_32,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_33,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_34,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_35,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_36,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_37,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_38,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_39,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_40,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_41,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_42,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_43,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_44,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_45,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_46,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_47,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_48,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_49,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_50,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_51,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_52,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_53,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom_54,axiom,
    normal8620421768224518004emidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_55,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_56,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_57,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_58,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_59,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_60,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_61,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_62,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_63,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_64,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_65,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_66,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_67,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_68,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_69,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_70,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_71,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_72,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_73,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_74,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_75,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_76,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_77,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_78,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_79,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_80,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_81,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_82,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_83,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_84,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_85,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_86,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_87,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_88,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_89,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_90,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_91,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_92,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_93,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_94,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Power_Opower_95,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_96,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_97,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_98,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_99,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_100,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_101,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_102,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_103,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_104,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_105,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_106,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_107,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_108,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_109,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_110,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_111,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_112,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_113,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_114,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_115,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_116,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_117,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_118,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_119,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_120,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_121,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_122,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_123,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_124,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_125,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_126,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_127,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_128,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_129,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_130,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_131,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_132,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_133,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_134,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_135,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_136,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_137,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_138,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_139,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_140,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_141,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_142,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_143,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_144,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_145,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_146,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_147,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_148,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_149,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_150,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_151,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_152,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_153,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_154,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_155,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_156,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_157,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_158,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_159,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_160,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_161,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_162,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_163,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_164,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_165,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_166,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_167,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_168,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_169,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_170,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_171,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_172,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_173,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_174,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_175,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_176,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_177,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_178,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_179,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_180,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_181,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_182,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_183,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_184,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_185,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_186,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_187,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_188,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_189,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_190,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_191,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_192,axiom,
    ! [A17: $tType] : condit1219197933456340205attice(set(A17)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_193,axiom,
    ! [A17: $tType] : counta3822494911875563373attice(set(A17)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_194,axiom,
    ! [A17: $tType] : comple592849572758109894attice(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_195,axiom,
    ! [A17: $tType] : bounde4967611905675639751up_bot(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_196,axiom,
    ! [A17: $tType] : bounde4346867609351753570nf_top(set(A17)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_197,axiom,
    ! [A17: $tType] : comple6319245703460814977attice(set(A17)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_198,axiom,
    ! [A17: $tType] : boolea8198339166811842893lgebra(set(A17)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_199,axiom,
    ! [A17: $tType] : comple9053668089753744459l_ccpo(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_200,axiom,
    ! [A17: $tType] : semilattice_sup(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_201,axiom,
    ! [A17: $tType] : semilattice_inf(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_202,axiom,
    ! [A17: $tType] : order_top(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_203,axiom,
    ! [A17: $tType] : order_bot(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_204,axiom,
    ! [A17: $tType] : preorder(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Olattice_205,axiom,
    ! [A17: $tType] : lattice(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder_206,axiom,
    ! [A17: $tType] : order(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oord_207,axiom,
    ! [A17: $tType] : ord(set(A17)) ).

tff(tcon_Set_Oset___Groups_Ouminus_208,axiom,
    ! [A17: $tType] : uminus(set(A17)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_209,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_210,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_211,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_212,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_213,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_214,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_215,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_216,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_217,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_218,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_219,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_220,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_221,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_222,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_223,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_224,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_225,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_226,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_227,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_228,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_229,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_230,axiom,
    uminus($o) ).

tff(tcon_List_Olist___Nat_Osize_231,axiom,
    ! [A17: $tType] : size(list(A17)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_232,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_233,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_234,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_235,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_236,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_237,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_238,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_239,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_240,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_241,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_242,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_243,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_244,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_245,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_246,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
    real_V768167426530841204y_dist(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_247,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_248,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_249,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_250,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_251,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_252,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_253,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
    topolo7287701948861334536_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_254,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_255,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_256,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_257,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_258,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_259,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_260,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_261,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_262,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_263,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_264,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_265,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_266,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_267,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_268,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_269,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_270,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_271,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_272,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_273,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_274,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_275,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_276,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_277,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_278,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_279,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_280,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_281,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_282,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_283,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_284,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_285,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_286,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_287,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_288,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_289,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_290,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_291,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_292,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_293,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_294,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_295,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_296,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_297,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_298,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_299,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_300,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_301,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_302,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_303,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_304,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_305,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_306,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_307,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_308,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_309,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_310,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_311,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_312,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_313,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_314,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_315,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_316,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_317,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_318,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_319,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_320,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_321,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_322,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_323,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_324,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_325,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_326,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_327,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_328,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_329,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_330,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_331,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_332,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_333,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_334,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_335,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_336,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Nat_Osize_337,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_338,axiom,
    ! [A17: $tType,A18: $tType] : size(sum_sum(A17,A18)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_339,axiom,
    ! [A17: $tType] : condit1219197933456340205attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_340,axiom,
    ! [A17: $tType] : counta3822494911875563373attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_341,axiom,
    ! [A17: $tType] : bounde4967611905675639751up_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_342,axiom,
    ! [A17: $tType] : bounde4346867609351753570nf_top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_343,axiom,
    ! [A17: $tType] : comple6319245703460814977attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_344,axiom,
    ! [A17: $tType] : comple9053668089753744459l_ccpo(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_345,axiom,
    ! [A17: $tType] : semilattice_sup(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_346,axiom,
    ! [A17: $tType] : semilattice_inf(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_347,axiom,
    ! [A17: $tType] : order_top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_348,axiom,
    ! [A17: $tType] : order_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_349,axiom,
    ! [A17: $tType] : preorder(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_350,axiom,
    ! [A17: $tType] : lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_351,axiom,
    ! [A17: $tType] : order(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_352,axiom,
    ! [A17: $tType] : ord(filter(A17)) ).

tff(tcon_Option_Ooption___Nat_Osize_353,axiom,
    ! [A17: $tType] : size(option(A17)) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_354,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_355,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_356,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_357,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_358,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_359,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_360,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___Groups_Oplus_361,axiom,
    plus(literal) ).

tff(tcon_String_Oliteral___Nat_Osize_362,axiom,
    size(literal) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_363,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_364,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_365,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_366,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_367,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_368,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_369,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_370,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_371,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_372,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_373,axiom,
    real_V768167426530841204y_dist(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___Topological__Spaces_Ouniform__space_379,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_380,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_381,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_382,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_383,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_384,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_385,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_386,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_387,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_388,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_389,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_390,axiom,
    comm_s4317794764714335236cancel(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___Fields_Ofield_429,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_430,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_431,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_432,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_433,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_434,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_435,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_436,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_437,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_438,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_439,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_440,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_441,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_442,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_443,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_444,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_445,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_446,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_447,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_448,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_449,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_450,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_451,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_452,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_453,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_454,axiom,
    comple9053668089753744459l_ccpo(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___Groups_Oab__semigroup__mult_457,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_458,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_459,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_460,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_461,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_462,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_463,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_464,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_465,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_466,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_467,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_468,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_469,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_470,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_471,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_472,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_473,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_474,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_475,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_476,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_477,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_478,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_479,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_480,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_481,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_482,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_483,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_484,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_485,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_486,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_487,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_488,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_489,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_490,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_491,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( topolo4958980785337419405_space(A17)
        & topolo4958980785337419405_space(A18) )
     => topolo4958980785337419405_space(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_492,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( topological_t2_space(A17)
        & topological_t2_space(A18) )
     => topological_t2_space(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_493,axiom,
    ! [A17: $tType,A18: $tType] : size(product_prod(A17,A18)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_494,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_495,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_496,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_497,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_498,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_499,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_500,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_501,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_502,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_503,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_504,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_505,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_506,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_507,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_508,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_509,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_510,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_511,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_512,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_513,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_514,axiom,
    uminus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_515,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_516,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_517,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_518,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_519,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_520,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_521,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_522,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_523,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_524,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_525,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_526,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_527,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_528,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_529,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_530,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_531,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_532,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_533,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_534,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_535,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_536,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_537,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_538,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_539,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_540,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_541,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_542,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_543,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_544,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_545,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_546,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_547,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_548,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_549,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_550,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_551,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_552,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_553,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_554,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_555,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_556,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_557,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_558,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_559,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_560,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_561,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_562,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_563,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_564,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_565,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_566,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_567,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_568,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_569,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_570,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_571,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_572,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_573,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_574,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_575,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_576,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_577,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_578,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_579,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_580,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_581,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_582,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_583,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_584,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_585,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_586,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_587,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_588,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_589,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_590,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_591,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_592,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_593,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_594,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_595,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_596,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_597,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_598,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_599,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_600,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_601,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_602,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_603,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_604,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_605,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_606,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_607,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_608,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_609,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_610,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_611,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_612,axiom,
    dvd(code_integer) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_613,axiom,
    size(vEBT_VEBT) ).

% Helper facts (5)
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: $tType,X16: A,Y: A] :
      ( ( X16 != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X16),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X16: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X16),Y)
      | ( X16 = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X7: A] : aa(A,$o,P,X7) ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary),xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,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)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2))))))) ).

%------------------------------------------------------------------------------