%------------------------------------------------------------------------------
% File     : ITP226_2 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Member 00468_022002
% 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    : 0065_VEBT_Member_00468_022002 [Des22]

% Status   : Theorem
% Rating   : 0.50 v8.1.0
% Syntax   : Number of formulae    : 11555 (2848 unt;1635 typ;   0 def)
%            Number of atoms       : 27138 (8504 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 19267 (2049   ~; 317   |;2308   &)
%                                         (1966 <=>;12627  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   40 (   2 avg)
%            Number of types       :   15 (  14 usr)
%            Number of type conns  : 1397 (1119   >; 278   *;   0   +;   0  <<)
%            Number of predicates  :  270 ( 267 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1354 (1354 usr;  88 con; 0-8 aty)
%            Number of variables   : 29840 (26747   !; 644   ?;29840   :)
%                                         (2449  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 19:06:33.555
%------------------------------------------------------------------------------
% Could-be-implicit typings (22)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

tff(ty_t_Code__Numeral_Onatural,type,
    code_natural: $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_HOL_Obool,type,
    bool: $tType ).

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

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

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

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

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

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

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

% Explicit typings (1613)
tff(sy_cl_Lattices_Obounded__lattice,type,
    bounded_lattice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Real__Vector__Spaces_Odist__norm,type,
    real_V6936659425649961206t_norm: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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_Rings_Osemidom__divide__unit__factor,type,
    semido2269285787275462019factor: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Operfect__space,type,
    topolo8386298272705272623_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_Limits_Otopological__comm__monoid__add,type,
    topolo5987344860129210374id_add: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
    comple592849572758109894attice: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
    normal6328177297339901930cative: 
      !>[A: $tType] : $o ).

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

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

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

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

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

tff(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
    condit5016429287641298734tinuum: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
    counta3822494911875563373attice: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
    condit1219197933456340205attice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__jv____,type,
    aTP_Lamp_jv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jw____,type,
    aTP_Lamp_jw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jx____,type,
    aTP_Lamp_jx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jy____,type,
    aTP_Lamp_jy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jz____,type,
    aTP_Lamp_jz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ka____,type,
    aTP_Lamp_ka: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kb____,type,
    aTP_Lamp_kb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kc____,type,
    aTP_Lamp_kc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kd____,type,
    aTP_Lamp_kd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ke____,type,
    aTP_Lamp_ke: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kf____,type,
    aTP_Lamp_kf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kg____,type,
    aTP_Lamp_kg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kh____,type,
    aTP_Lamp_kh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ki____,type,
    aTP_Lamp_ki: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

tff(sy_c_ATP_058Lamp__kl____,type,
    aTP_Lamp_kl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__km____,type,
    aTP_Lamp_km: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kn____,type,
    aTP_Lamp_kn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__lg____,type,
    aTP_Lamp_lg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lh____,type,
    aTP_Lamp_lh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__li____,type,
    aTP_Lamp_li: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__nv____,type,
    aTP_Lamp_nv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

tff(sy_c_ATP_058Lamp__om____,type,
    aTP_Lamp_om: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__on____,type,
    aTP_Lamp_on: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

tff(sy_c_ATP_058Lamp__os____,type,
    aTP_Lamp_os: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ot____,type,
    aTP_Lamp_ot: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ou____,type,
    aTP_Lamp_ou: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__pi____,type,
    aTP_Lamp_pi: 
      !>[A: $tType,B: $tType] : fun(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] : ( 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] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qw____,type,
    aTP_Lamp_qw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qx____,type,
    aTP_Lamp_qx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qy____,type,
    aTP_Lamp_qy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__sp____,type,
    aTP_Lamp_sp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sq____,type,
    aTP_Lamp_sq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sr____,type,
    aTP_Lamp_sr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ss____,type,
    aTP_Lamp_ss: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__st____,type,
    aTP_Lamp_st: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__su____,type,
    aTP_Lamp_su: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sv____,type,
    aTP_Lamp_sv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sw____,type,
    aTP_Lamp_sw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sx____,type,
    aTP_Lamp_sx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sy____,type,
    aTP_Lamp_sy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sz____,type,
    aTP_Lamp_sz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ta____,type,
    aTP_Lamp_ta: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tb____,type,
    aTP_Lamp_tb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tc____,type,
    aTP_Lamp_tc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__td____,type,
    aTP_Lamp_td: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

tff(sy_c_ATP_058Lamp__tj____,type,
    aTP_Lamp_tj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tk____,type,
    aTP_Lamp_tk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tl____,type,
    aTP_Lamp_tl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tm____,type,
    aTP_Lamp_tm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tn____,type,
    aTP_Lamp_tn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__vq____,type,
    aTP_Lamp_vq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__xd____,type,
    aTP_Lamp_xd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xe____,type,
    aTP_Lamp_xe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : ( 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_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__Def_Orel__fun,type,
    bNF_rel_fun: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(fun(A,B),fun(fun(C,D),bool)) ) ).

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_Odir__image,type,
    bNF_We2720479622203943262_image: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * fun(A,A2) ) > set(product_prod(A2,A2)) ) ).

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_Omax2,type,
    bNF_We1388413361240627857o_max2: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A * 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__BNFs_Orel__prod,type,
    basic_rel_prod: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,bool)) * fun(C,fun(D,bool)) ) > fun(product_prod(A,C),fun(product_prod(B,D),bool)) ) ).

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),bool)) ).

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),bool)) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int * int ) > int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

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

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

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

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

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

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,bool) ) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > bool ) ).

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

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: ( num * 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),bool)) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: ( num * 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),bool)) ).

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_COMBB,type,
    combb: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).

tff(sy_c_COMBC,type,
    combc: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).

tff(sy_c_COMBS,type,
    combs: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).

tff(sy_c_Code__Numeral_OSuc,type,
    code_Suc: fun(code_natural,code_natural) ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,bool) ).

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: fun(code_integer,int) ).

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

tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: fun(nat,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger__of__natural,type,
    code_i5400310926305786745atural: fun(code_natural,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_Onatural_Onat__of__natural,type,
    code_nat_of_natural: fun(code_natural,nat) ).

tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: fun(nat,code_natural) ).

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun(num,code_integer) ).

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

tff(sy_c_Code__Numeral_Opcr__integer,type,
    code_pcr_integer: fun(int,fun(code_integer,bool)) ).

tff(sy_c_Code__Numeral_Opcr__natural,type,
    code_pcr_natural: fun(nat,fun(code_natural,bool)) ).

tff(sy_c_Code__Numeral_Osub,type,
    code_sub: fun(num,fun(num,code_integer)) ).

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

tff(sy_c_Code__Target__Nat_Oint__of__nat,type,
    code_T6385005292777649522of_nat: fun(nat,int) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_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_Conditionally__Complete__Lattices_Opreordering__bdd,type,
    condit622319405099724424ng_bdd: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

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

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

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

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

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: ( int * int ) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

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

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

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

tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
    euclid6346220572633701492n_size: 
      !>[A: $tType] : ( 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: extended_enat > extended_enat ).

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

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

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

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

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

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

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,bool) * filter(A) ) > $o ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Fun_Oid,type,
    id: 
      !>[A: $tType] : fun(A,A) ).

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

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

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

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

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

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

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

tff(sy_c_GCD_OGcd__class_OLcm,type,
    gcd_Lcm: 
      !>[A: $tType] : ( set(A) > A ) ).

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

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

tff(sy_c_GCD_Obounded__quasi__semilattice,type,
    bounde8507323023520639062attice: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set,type,
    bounde6485984586167503788ce_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $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),bool)) ).

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

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

tff(sy_c_Groups_Ogroup__axioms,type,
    group_axioms: 
      !>[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_Omonoid__axioms,type,
    monoid_axioms: 
      !>[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_Osemigroup,type,
    semigroup: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

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

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

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

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

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

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

tff(sy_c_Groups__List_Ocomm__monoid__list,type,
    groups1828464146339083142d_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Ocomm__monoid__list__set,type,
    groups4802862169904069756st_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

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,bool) > A ) ).

tff(sy_c_HOL_OUniq,type,
    uniq: 
      !>[A: $tType] : ( fun(A,bool) > $o ) ).

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

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

tff(sy_c_If,type,
    if: 
      !>[A: $tType] : ( ( bool * A * A ) > 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_ONeg,type,
    neg: num > int ).

tff(sy_c_Int_OPos,type,
    pos: fun(num,int) ).

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

tff(sy_c_Int_Ocr__int,type,
    cr_int: fun(product_prod(nat,nat),fun(int,bool)) ).

tff(sy_c_Int_Odup,type,
    dup: int > int ).

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),bool)) ).

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,bool)) ).

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_Int_Osub,type,
    sub: ( num * num ) > int ).

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,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices_Osemilattice__order,type,
    semilattice_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
    lattices_ord_arg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > 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_Oord__class_Ois__arg__max,type,
    lattic501386751176901750rg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) * B ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
    lattic5652469242046573047tr_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
    lattic5214292709420241887eutr_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * 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,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__set,type,
    lattic149705377957585745ce_set: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__set_OF,type,
    lattic1715443433743089157tice_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).

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,bool)) * fun(A,B) * fun(B,A) * fun(A,fun(B,bool)) ) > $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_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_Ocoset,type,
    coset: 
      !>[A: $tType] : ( list(A) > set(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,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(A) ) ).

tff(sy_c_List_Ofold,type,
    fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).

tff(sy_c_List_Ofolding__insort__key,type,
    folding_insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * set(B) * fun(B,A) ) > $o ) ).

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_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,A) * set(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_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__all,type,
    list_all: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

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,bool) * 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_Omap__filter,type,
    map_filter: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(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),bool)) ).

tff(sy_c_List_Oord__class_Olexordp__eq,type,
    ord_lexordp_eq: 
      !>[A: $tType] : ( ( list(A) * 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,bool)) * 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,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Othose,type,
    those: 
      !>[A: $tType] : ( list(option(A)) > option(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_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

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__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__le,type,
    map_le: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > $o ) ).

tff(sy_c_Map_Omap__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__upds,type,
    map_upds: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Ocompow,type,
    compow: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Nat_Ofunpow,type,
    funpow: 
      !>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,bool) ) ).

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_Oint__decode,type,
    nat_int_decode: fun(nat,int) ).

tff(sy_c_Nat__Bijection_Oint__encode,type,
    nat_int_encode: fun(int,nat) ).

tff(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: fun(nat,list(nat)) ).

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: fun(list(nat),nat) ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),bool)) ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun(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),bool)) ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),nat) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: nat > fun(real,real) ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun(real,real) ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Onat__of__num,type,
    nat_of_num: fun(num,nat) ).

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_Ois__num,type,
    neg_numeral_is_num: 
      !>[A: $tType] : ( A > $o ) ).

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_Ois__none,type,
    is_none: 
      !>[A: $tType] : ( option(A) > $o ) ).

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,bool),fun(option(A),bool)) ).

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,bool)),fun(option(A),fun(option(B),bool))) ).

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__Relation_Olinear__order__on,type,
    order_679001287576687338der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,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_Ostrict__linear__order__on,type,
    order_5396836661320670305der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

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__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

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_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
    order_strict_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oordering__top,type,
    ordering_top: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * A ) > $o ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Partial__Function_Oflat__lub,type,
    partial_flat_lub: 
      !>[A: $tType] : ( ( A * set(A) ) > A ) ).

tff(sy_c_Power_Opower_Opower,type,
    power2: 
      !>[A: $tType] : ( ( A * fun(A,fun(A,A)) * A * 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] : ( 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_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).

tff(sy_c_Product__Type_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_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).

tff(sy_c_Product__Type_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_Oscomp,type,
    product_scomp: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Quotient_OQuotient3,type,
    quotient3: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,B) * fun(B,A) ) > $o ) ).

tff(sy_c_Random_Oinc__shift,type,
    inc_shift: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Oiterate,type,
    iterate: 
      !>[B: $tType,A: $tType] : ( ( code_natural * fun(B,fun(A,product_prod(B,A))) ) > fun(B,fun(A,product_prod(B,A))) ) ).

tff(sy_c_Random_Oiterate__rel,type,
    iterate_rel: 
      !>[B: $tType,A: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool)) ).

tff(sy_c_Random_Olog,type,
    log: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Olog__rel,type,
    log_rel: fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),bool)) ).

tff(sy_c_Random_Ominus__shift,type,
    minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Onext,type,
    next: fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Opick,type,
    pick: 
      !>[A: $tType] : ( ( list(product_prod(code_natural,A)) * code_natural ) > A ) ).

tff(sy_c_Random_Orange,type,
    range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Oselect,type,
    select: 
      !>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random_Oselect__weight,type,
    select_weight: 
      !>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun(product_prod(int,int),rat) ).

tff(sy_c_Rat_OFract,type,
    fract: fun(int,fun(int,rat)) ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod(int,int) > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun(rat,product_prod(int,int)) ).

tff(sy_c_Rat_Ofield__char__0__class_ORats,type,
    field_char_0_Rats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
    field_char_0_of_rat: 
      !>[A: $tType] : fun(rat,A) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,bool)) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,bool) ).

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),bool)) ).

tff(sy_c_Real_ORatreal,type,
    ratreal: fun(rat,real) ).

tff(sy_c_Real_OReal,type,
    real2: fun(fun(nat,rat),real) ).

tff(sy_c_Real_Ocauchy,type,
    cauchy: fun(nat,rat) > $o ).

tff(sy_c_Real_Ocr__real,type,
    cr_real: fun(fun(nat,rat),fun(real,bool)) ).

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun(fun(nat,rat),fun(real,bool)) ).

tff(sy_c_Real_Opositive,type,
    positive2: fun(real,bool) ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun(fun(nat,rat),fun(fun(nat,rat),bool)) ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun(real,fun(nat,rat)) ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun(nat,rat) > bool ).

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_Oconstruct,type,
    real_V4425403222259421789struct: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * A ) > B ) ).

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_Oextend__basis,type,
    real_V4986007116245087402_basis: 
      !>[A: $tType] : ( set(A) > set(A) ) ).

tff(sy_c_Real__Vector__Spaces_Olinear,type,
    real_Vector_linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

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 * A ) > A ) ).

tff(sy_c_Real__Vector__Spaces_Ospan,type,
    real_Vector_span: 
      !>[A: $tType] : ( set(A) > set(A) ) ).

tff(sy_c_Real__Vector__Spaces_Osubspace,type,
    real_Vector_subspace: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Relation_ODomain,type,
    domain: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(A) ) ).

tff(sy_c_Relation_OField,type,
    field2: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).

tff(sy_c_Relation_OId,type,
    id2: 
      !>[A: $tType] : set(product_prod(A,A)) ).

tff(sy_c_Relation_ORange,type,
    range2: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(B) ) ).

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_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Orelcomp,type,
    relcomp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).

tff(sy_c_Relation_Ototal__on,type,
    total_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Otransp,type,
    transp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $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] : fun(A,fun(A,bool)) ).

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] : fun(A,A) ).

tff(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
    unit_f5069060285200089521factor: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun(bool,A) ).

tff(sy_c_Series_Osuminf,type,
    suminf: 
      !>[A: $tType] : ( fun(nat,A) > A ) ).

tff(sy_c_Series_Osummable,type,
    summable: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( ( fun(nat,A) * A ) > $o ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : ( fun(A,bool) > set(A) ) ).

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Oimage,type,
    image: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > set(B) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : ( A > fun(set(A),set(A)) ) ).

tff(sy_c_Set_Ois__singleton,type,
    is_singleton: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Opairwise,type,
    pairwise: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).

tff(sy_c_Set_Oremove,type,
    remove: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Ovimage,type,
    vimage: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(B) ) > set(A) ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_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))),bool)) ).

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: ( bool * bool * bool * bool * bool * bool * bool * literal ) > literal ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : fun(A,char) ).

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) ) > fun(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_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_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] : fun(A,A) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun(nat,real) ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( fun(nat,A) > fun(nat,A) ) ).

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log2: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Transcendental_Opowr__real,type,
    powr_real: fun(real,fun(real,real)) ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun(nat,real) ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transitive__Closure_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_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( bool * bool ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option(product_prod(nat,nat)) * nat * list(vEBT_VEBT) * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT,type,
    vEBT_case_VEBT: 
      !>[A: $tType] : fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A))) ).

tff(sy_c_VEBT__Definitions_OVEBT_Orec__VEBT,type,
    vEBT_rec_VEBT: 
      !>[A: $tType] : fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(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: fun(vEBT_VEBT,fun(nat,bool)) ).

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),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__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,bool)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,bool) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).

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_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Owf,type,
    wf: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : ( fun(A,bool) > bool ) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_fEx,type,
    fEx: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun(bool,bool) ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: ( bool * bool ) > bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_fimplies,type,
    fimplies: fun(bool,fun(bool,bool)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : fun(A,fun(set(A),bool)) ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_n,type,
    n: nat ).

tff(sy_v_t,type,
    t: vEBT_VEBT ).

tff(sy_v_x,type,
    x: nat ).

% Relevant facts (9010)
tff(fact_0_min__Null__member,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_minNull(T2)
     => ~ pp(aa(nat,bool,vEBT_vebt_member(T2),X)) ) ).

% min_Null_member
tff(fact_1_valid__eq,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
    <=> vEBT_invar_vebt(T2,D2) ) ).

% valid_eq
tff(fact_2_valid__eq1,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(T2,D2)
     => vEBT_VEBT_valid(T2,D2) ) ).

% valid_eq1
tff(fact_3_valid__eq2,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
     => vEBT_invar_vebt(T2,D2) ) ).

% valid_eq2
tff(fact_4_both__member__options__equiv__member,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
      <=> pp(aa(nat,bool,vEBT_vebt_member(T2),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_5_valid__member__both__member__options,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
       => pp(aa(nat,bool,vEBT_vebt_member(T2),X)) ) ) ).

% valid_member_both_member_options
tff(fact_6_valid__0__not,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% valid_0_not
tff(fact_7_valid__tree__deg__neq__0,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_8_deg__deg__n,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),N)
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_9_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(Tree,N)
     => ( pp(aa(nat,bool,vEBT_vebt_member(Tree),X))
       => ( vEBT_V5719532721284313246member(Tree,X)
          | vEBT_VEBT_membermima(Tree,X) ) ) ) ).

% member_valid_both_member_options
tff(fact_10_deg__not__0,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% deg_not_0
tff(fact_11_Leaf__0__not,axiom,
    ! [A3: bool,B2: bool] : ~ vEBT_invar_vebt(vEBT_Leaf(A3,B2),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_12_not__min__Null__member,axiom,
    ! [T2: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(T2)
     => ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X_1)) ) ).

% not_min_Null_member
tff(fact_13_both__member__options__def,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
    <=> ( vEBT_V5719532721284313246member(T2,X)
        | vEBT_VEBT_membermima(T2,X) ) ) ).

% both_member_options_def
tff(fact_14_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ vEBT_VEBT_minNull(vEBT_Leaf(Uu,fTrue)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_15_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ vEBT_VEBT_minNull(vEBT_Leaf(fTrue,Uv)) ).

% VEBT_internal.minNull.simps(2)
tff(fact_16_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull(vEBT_Leaf(fFalse,fFalse)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_17_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => vEBT_invar_vebt(vEBT_vebt_buildup(N),N) ) ).

% buildup_gives_valid
tff(fact_18_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A3)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_19_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% neq0_conv
tff(fact_20_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_nat_zero_code
tff(fact_21_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_22_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_leaf
tff(fact_23_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_24_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_min_max
tff(fact_25_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X22: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X22) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X22)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_26_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_27_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_28_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_29_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_30_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A3: A] :
          ( ! [X2: A] :
              ( ! [Y: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,Y)),aa(A,B,F2,X2)))
                 => pp(aa(A,bool,P,Y)) )
             => pp(aa(A,bool,P,X2)) )
         => pp(aa(A,bool,P,A3)) ) ) ).

% measure_induct_rule
tff(fact_31_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A3: A] :
          ( ! [X2: A] :
              ( ! [Y: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,Y)),aa(A,B,F2,X2)))
                 => pp(aa(A,bool,P,Y)) )
             => pp(aa(A,bool,P,X2)) )
         => pp(aa(A,bool,P,A3)) ) ) ).

% measure_induct
tff(fact_32_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,bool),V: fun(A,nat),X: A] :
      ( ! [X2: A] :
          ( ~ pp(aa(A,bool,P,X2))
         => ? [Y: A] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y)),aa(A,nat,V,X2)))
              & ~ pp(aa(A,bool,P,Y)) ) )
     => pp(aa(A,bool,P,X)) ) ).

% infinite_descent_measure
tff(fact_33_linorder__neqE__nat,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X != Y2 )
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y2),X)) ) ) ).

% linorder_neqE_nat
tff(fact_34_infinite__descent,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa(nat,bool,P,N2))
         => ? [M: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
              & ~ pp(aa(nat,bool,P,M)) ) )
     => pp(aa(nat,bool,P,N)) ) ).

% infinite_descent
tff(fact_35_nat__less__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
             => pp(aa(nat,bool,P,M)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% nat_less_induct
tff(fact_36_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_irrefl_nat
tff(fact_37_less__not__refl3,axiom,
    ! [S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
     => ( S != T2 ) ) ).

% less_not_refl3
tff(fact_38_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( M2 != N ) ) ).

% less_not_refl2
tff(fact_39_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_not_refl
tff(fact_40_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ).

% nat_neq_iff
tff(fact_41_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_42_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M2: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),N))
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_43_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),collect(A,P)))
    <=> pp(aa(A,bool,P,A3)) ) ).

% mem_Collect_eq
tff(fact_44_Collect__mem__eq,axiom,
    ! [A: $tType,A4: set(A)] : collect(A,aTP_Lamp_a(set(A),fun(A,bool),A4)) = A4 ).

% Collect_mem_eq
tff(fact_45_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X2: A] :
          ( pp(aa(A,bool,P,X2))
        <=> pp(aa(A,bool,Q,X2)) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_46_ext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X2: A] : aa(A,B,F2,X2) = aa(A,B,G,X2)
     => ( F2 = G ) ) ).

% ext
tff(fact_47_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N),zero_zero(A))) ) ).

% not_less_zero
tff(fact_48_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ( N != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N)) ) ) ).

% gr_zeroI
tff(fact_49_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P: fun(A,bool),X: A] :
      ( ! [X2: A] :
          ( ( aa(A,nat,V,X2) = zero_zero(nat) )
         => pp(aa(A,bool,P,X2)) )
     => ( ! [X2: A] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X2)))
           => ( ~ pp(aa(A,bool,P,X2))
             => ? [Y: A] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y)),aa(A,nat,V,X2)))
                  & ~ pp(aa(A,bool,P,Y)) ) ) )
       => pp(aa(A,bool,P,X)) ) ) ).

% infinite_descent0_measure
tff(fact_50_infinite__descent0,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( ~ pp(aa(nat,bool,P,N2))
             => ? [M: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
                  & ~ pp(aa(nat,bool,P,M)) ) ) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% infinite_descent0
tff(fact_51_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_52_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_zeroE
tff(fact_53_not__less0,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% not_less0
tff(fact_54_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_55_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% gr0I
tff(fact_56_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),zero_zero(nat))) ).

% bot_nat_0.extremum_strict
tff(fact_57_VEBT_Oexhaust,axiom,
    ! [Y2: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Y2 != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: bool,X222: bool] : Y2 != vEBT_Leaf(X212,X222) ) ).

% VEBT.exhaust
tff(fact_58_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: bool,X22: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X22) ).

% VEBT.distinct(1)
tff(fact_59_deg1Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,one_one(nat))
    <=> ? [A5: bool,B3: bool] : T2 = vEBT_Leaf(A5,B3) ) ).

% deg1Leaf
tff(fact_60_deg__1__Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,one_one(nat))
     => ? [A6: bool,B4: bool] : T2 = vEBT_Leaf(A6,B4) ) ).

% deg_1_Leaf
tff(fact_61_deg__1__Leafy,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( N = one_one(nat) )
       => ? [A6: bool,B4: bool] : T2 = vEBT_Leaf(A6,B4) ) ) ).

% deg_1_Leafy
tff(fact_62_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D1))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D22))
           => ? [E: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),E),D1))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),E),D22)) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_63_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),zero_zero(A))) ) ).

% less_numeral_extra(3)
tff(fact_64_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,N)))
     => ? [Info2: option(product_prod(nat,nat)),TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,N)),TreeList2,S2) ) ).

% deg_SUcn_Node
tff(fact_65_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X22: bool] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf(X21,X22)) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_66_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X22: bool] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf(X21,X22)) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_67_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% of_nat_0_less_iff
tff(fact_68_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A3: bool,B2: bool,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf(A3,B2),X)
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A3) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_69_vebt__member_Osimps_I1_J,axiom,
    ! [A3: bool,B2: bool,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Leaf(A3,B2)),X))
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A3) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_70_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(X)
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_71_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_72_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( aa(nat,nat,suc,X23) = aa(nat,nat,suc,Y23) )
    <=> ( X23 = Y23 ) ) ).

% nat.inject
tff(fact_73_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M2: nat,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M2) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( M2 = N ) ) ) ).

% of_nat_eq_iff
tff(fact_74_lessI,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,N))) ).

% lessI
tff(fact_75_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N))) ) ).

% Suc_mono
tff(fact_76_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_less_eq
tff(fact_77_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_78_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( zero_zero(nat) = N ) ) ) ).

% of_nat_0_eq_iff
tff(fact_79_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M2) = zero_zero(A) )
        <=> ( M2 = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_80_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% of_nat_less_iff
tff(fact_81_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_82_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N))) ).

% zero_less_Suc
tff(fact_83_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_84_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_85_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_86_less__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),one_one(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_87_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_88_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_89_Suc__inject,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y2) )
     => ( X = Y2 ) ) ).

% Suc_inject
tff(fact_90_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_91_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_92_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),one_one(A))) ) ).

% less_numeral_extra(4)
tff(fact_93_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y2) )
         => ( X != Y2 ) ) ) ).

% size_neq_size_imp_neq
tff(fact_94_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_95_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero(nat) != aa(nat,nat,suc,X23) ).

% nat.distinct(1)
tff(fact_96_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_97_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_98_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa(nat,nat,suc,X23) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_99_old_Onat_Oexhaust,axiom,
    ! [Y2: nat] :
      ( ( Y2 != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y2 != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_100_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ( ( X != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va: nat] : X != aa(nat,nat,suc,aa(nat,nat,suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_101_nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% nat_induct
tff(fact_102_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M2: nat,N: nat] :
      ( ! [X2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X2),zero_zero(nat)))
     => ( ! [Y3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,zero_zero(nat)),aa(nat,nat,suc,Y3)))
       => ( ! [X2: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X2),Y3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,aa(nat,nat,suc,X2)),aa(nat,nat,suc,Y3))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M2),N)) ) ) ) ).

% diff_induct
tff(fact_103_zero__induct,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ).

% zero_induct
tff(fact_104_Suc__neq__Zero,axiom,
    ! [M2: nat] : aa(nat,nat,suc,M2) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_105_Zero__neq__Suc,axiom,
    ! [M2: nat] : zero_zero(nat) != aa(nat,nat,suc,M2) ).

% Zero_neq_Suc
tff(fact_106_Zero__not__Suc,axiom,
    ! [M2: nat] : zero_zero(nat) != aa(nat,nat,suc,M2) ).

% Zero_not_Suc
tff(fact_107_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M3: nat] : N = aa(nat,nat,suc,M3) ) ).

% not0_implies_Suc
tff(fact_108_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K))
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
             => ( K != aa(nat,nat,suc,J) ) ) ) ) ).

% Nat.lessE
tff(fact_109_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_lessD
tff(fact_110_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),K))
     => ~ ! [J: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
           => ( K != aa(nat,nat,suc,J) ) ) ) ).

% Suc_lessE
tff(fact_111_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( ( aa(nat,nat,suc,M2) != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),N)) ) ) ).

% Suc_lessI
tff(fact_112_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => ( M2 = N ) ) ) ).

% less_SucE
tff(fact_113_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N))) ) ).

% less_SucI
tff(fact_114_Ex__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P,I2)) )
    <=> ( pp(aa(nat,bool,P,N))
        | ? [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
            & pp(aa(nat,bool,P,I2)) ) ) ) ).

% Ex_less_Suc
tff(fact_115_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | ( M2 = N ) ) ) ).

% less_Suc_eq
tff(fact_116_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2))) ) ).

% not_less_eq
tff(fact_117_All__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P,I2)) )
    <=> ( pp(aa(nat,bool,P,N))
        & ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
           => pp(aa(nat,bool,P,I2)) ) ) ) ).

% All_less_Suc
tff(fact_118_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
    <=> ? [M4: nat] :
          ( ( M2 = aa(nat,nat,suc,M4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M4)) ) ) ).

% Suc_less_eq2
tff(fact_119_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2)))
       => ( M2 = N ) ) ) ).

% less_antisym
tff(fact_120_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_less_SucD
tff(fact_121_less__trans__Suc,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_122_less__Suc__induct,axiom,
    ! [I: nat,J2: nat,P: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ( ! [I3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I3),aa(nat,nat,suc,I3)))
       => ( ! [I3: nat,J: nat,K2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K2))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I3),J))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,J),K2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I3),K2)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I),J2)) ) ) ) ).

% less_Suc_induct
tff(fact_123_strict__inc__induct,axiom,
    ! [I: nat,J2: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ( ! [I3: nat] :
            ( ( J2 = aa(nat,nat,suc,I3) )
           => pp(aa(nat,bool,P,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
             => ( pp(aa(nat,bool,P,aa(nat,nat,suc,I3)))
               => pp(aa(nat,bool,P,I3)) ) )
         => pp(aa(nat,bool,P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_124_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2)))
      <=> ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
tff(fact_125_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% less_numeral_extra(1)
tff(fact_126_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),zero_zero(A))) ) ).

% of_nat_less_0_iff
tff(fact_127_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) ) ) ).

% less_imp_of_nat_less
tff(fact_128_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% of_nat_less_imp_less
tff(fact_129_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N3: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,N3))) ) ) ) ).

% lift_Suc_mono_less
tff(fact_130_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,M2: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,M2)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_131_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P,I2)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        | ? [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,I2))) ) ) ) ).

% Ex_less_Suc2
tff(fact_132_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ? [M5: nat] : N = aa(nat,nat,suc,M5) ) ).

% gr0_conv_Suc
tff(fact_133_All__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P,I2)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        & ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,I2))) ) ) ) ).

% All_less_Suc2
tff(fact_134_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ? [M3: nat] : N = aa(nat,nat,suc,M3) ) ).

% gr0_implies_Suc
tff(fact_135_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N)))
    <=> ( ( M2 = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M2 = aa(nat,nat,suc,J3) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_136_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_137_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy2: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy2)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_138_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D2: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv),D2)
    <=> ( D2 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_139_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz: nat] : ~ vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy2),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_140_invar__vebt_Ointros_I1_J,axiom,
    ! [A3: bool,B2: bool] : vEBT_invar_vebt(vEBT_Leaf(A3,B2),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_141_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_142_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one
tff(fact_143_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) ) ).

% not_one_less_zero
tff(fact_144_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ~ ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% pos_int_cases
tff(fact_145_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ? [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
          & ( K = aa(nat,int,semiring_1_of_nat(int),N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_146_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N2: nat] : X != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_147_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,bool)),Q: fun(nat,fun(A,fun(A,bool)))] :
      ( ? [X_12: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,zero_zero(nat)),X_12))
     => ( ! [X2: A,N2: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N2),X2))
           => ? [Y: A] :
                ( pp(aa(A,bool,aa(nat,fun(A,bool),P,aa(nat,nat,suc,N2)),Y))
                & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N2),X2),Y)) ) )
       => ? [F3: fun(nat,A)] :
          ! [N4: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N4),aa(nat,A,F3,N4)))
            & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N4),aa(nat,A,F3,N4)),aa(nat,A,F3,aa(nat,nat,suc,N4)))) ) ) ) ).

% dependent_nat_choice
tff(fact_148_exists__least__lemma,axiom,
    ! [P: fun(nat,bool)] :
      ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
       => ? [N2: nat] :
            ( ~ pp(aa(nat,bool,P,N2))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_149_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% reals_Archimedean2
tff(fact_150_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_151_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_152_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = N ) ) ).

% Suc_diff_1
tff(fact_153_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_154_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_155_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_156_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_157_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_158_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_159_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ).

% diff_Suc_Suc
tff(fact_160_Suc__diff__diff,axiom,
    ! [M2: nat,N: 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,M2)),N)),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),M2),N)),K) ).

% Suc_diff_diff
tff(fact_161_diff__self__eq__0,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),M2) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_162_diff__0__eq__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_163_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% diff_gt_0_iff_gt
tff(fact_164_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_165_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% zero_less_diff
tff(fact_166_diff__Suc__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),one_one(nat)) = N ).

% diff_Suc_1
tff(fact_167_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% Suc_pred
tff(fact_168_int__int__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M2) = aa(nat,int,semiring_1_of_nat(int),N) )
    <=> ( M2 = N ) ) ).

% int_int_eq
tff(fact_169_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_170_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_171_diff__commute,axiom,
    ! [I: nat,J2: 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),J2)),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)),J2) ).

% diff_commute
tff(fact_172_less__int__code_I1_J,axiom,
    ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),zero_zero(int))) ).

% less_int_code(1)
tff(fact_173_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_174_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_175_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_176_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) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2)) ) ) ) ).

% diff_eq_diff_less
tff(fact_177_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),D2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_178_zero__induct__lemma,axiom,
    ! [P: fun(nat,bool),K: nat,I: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I))) ) ) ).

% zero_induct_lemma
tff(fact_179_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2) = zero_zero(nat) )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
tff(fact_180_minus__nat_Odiff__0,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),zero_zero(nat)) = M2 ).

% minus_nat.diff_0
tff(fact_181_less__imp__diff__less,axiom,
    ! [J2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),N)),K)) ) ).

% less_imp_diff_less
tff(fact_182_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M2))) ) ) ).

% diff_less_mono2
tff(fact_183_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_184_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ) ) ).

% Suc_diff_Suc
tff(fact_185_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),aa(nat,nat,suc,M2))) ).

% diff_less_Suc
tff(fact_186_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),M2)) ) ) ).

% diff_less
tff(fact_187_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_188_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_189_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] :
          ( ( X != Y2 )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_190_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_191_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_192_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,N: nat] :
      ( ! [N2: nat] : aa(nat,A,F2,aa(nat,nat,suc,N2)) = aa(nat,A,H,N2)
     => ( ( aa(nat,A,F2,zero_zero(nat)) = G )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,F2,N) = G ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,F2,N) = aa(nat,A,H,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ) ) ).

% Suc_if_eq
tff(fact_193_artanh__0,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ( aa(A,A,artanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% artanh_0
tff(fact_194_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arsinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_195_nat__int__comparison_I2_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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_196_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_197_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_198_minus__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A4: fun(A,B),B5: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A4),B5),X) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A4,X)),aa(A,B,B5,X)) ) ).

% minus_apply
tff(fact_199_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_200_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_201_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ~ ! [N2: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% neg_int_cases
tff(fact_202_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
         => ~ ! [N2: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),E2)) ) ) ).

% nat_approx_posE
tff(fact_203_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_204_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_205_verit__minus__simplify_I4_J,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B] : aa(B,B,uminus_uminus(B),aa(B,B,uminus_uminus(B),B2)) = B2 ) ).

% verit_minus_simplify(4)
tff(fact_206_uminus__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A4: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A4),X) = aa(B,B,uminus_uminus(B),aa(A,B,A4,X)) ) ).

% uminus_apply
tff(fact_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% neg_less_iff_less
tff(fact_215_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_216_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_217_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% less_neg_neg
tff(fact_218_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% neg_less_pos
tff(fact_219_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% neg_0_less_iff_less
tff(fact_220_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% neg_less_0_iff_less
tff(fact_221_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_222_verit__minus__simplify_I3_J,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),zero_zero(B)),B2) = aa(B,B,uminus_uminus(B),B2) ) ).

% verit_minus_simplify(3)
tff(fact_223_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_224_negative__eq__positive,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),M2) )
    <=> ( ( N = zero_zero(nat) )
        & ( M2 = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_225_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_226_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_227_negative__zless,axiom,
    ! [N: nat,M2: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),M2))) ).

% negative_zless
tff(fact_228_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_229_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_230_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_231_fun__Compl__def,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A4: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A4),X3) = aa(B,B,uminus_uminus(B),aa(A,B,A4,X3)) ) ).

% fun_Compl_def
tff(fact_232_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_233_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% verit_negate_coefficient(2)
tff(fact_234_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% less_minus_iff
tff(fact_235_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).

% minus_less_iff
tff(fact_236_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_237_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_238_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_239_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_240_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_241_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I3),K))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_less_induct
tff(fact_242_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M3: nat,N2: nat] : Z != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M3)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_diff_cases
tff(fact_243_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% int_cases2
tff(fact_244_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_245_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(4)
tff(fact_246_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% less_minus_one_simps(2)
tff(fact_247_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% int_cases
tff(fact_248_int__of__nat__induct,axiom,
    ! [P: fun(int,bool),Z: int] :
      ( ! [N2: nat] : pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),N2)))
     => ( ! [N2: nat] : pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2)))))
       => pp(aa(int,bool,P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_249_not__int__zless__negative,axiom,
    ! [N: nat,M2: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M2)))) ).

% not_int_zless_negative
tff(fact_250_int__ops_I6_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = zero_zero(int) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = 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_251_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(3)
tff(fact_252_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% less_minus_one_simps(1)
tff(fact_253_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N2: nat] : M2 != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( M2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ) ) ).

% int_cases4
tff(fact_254_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).

% verit_comp_simplify1(1)
tff(fact_255_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A4: fun(A,B),B5: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A4),B5),X3) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A4,X3)),aa(A,B,B5,X3)) ) ).

% fun_diff_def
tff(fact_256_option_Osize__neq,axiom,
    ! [A: $tType,X: option(A)] : aa(option(A),nat,size_size(option(A)),X) != zero_zero(nat) ).

% option.size_neq
tff(fact_257_int__if,axiom,
    ! [P: bool,A3: nat,B2: nat] :
      ( ( pp(P)
       => ( aa(nat,int,semiring_1_of_nat(int),if(nat,P,A3,B2)) = aa(nat,int,semiring_1_of_nat(int),A3) ) )
      & ( ~ pp(P)
       => ( aa(nat,int,semiring_1_of_nat(int),if(nat,P,A3,B2)) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ) ).

% int_if
tff(fact_258_nat__int__comparison_I1_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 = B2 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A3) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).

% nat_int_comparison(1)
tff(fact_259_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
             => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ) ).

% int_cases3
tff(fact_260_negD,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
     => ? [N2: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% negD
tff(fact_261_negative__zless__0,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int))) ).

% negative_zless_0
tff(fact_262_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% divide_less_0_1_iff
tff(fact_263_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_264_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_265_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_266_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_267_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% zero_less_divide_1_iff
tff(fact_268_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),X) ) ).

% divide_minus1
tff(fact_269_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_270_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_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_div__by__Suc__0,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),aa(nat,nat,suc,zero_zero(nat))) = M2 ).

% div_by_Suc_0
tff(fact_279_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_280_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_281_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_282_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_283_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_284_divide__self__if,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) = zero_zero(A) ) )
          & ( ( A3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ) ).

% divide_self_if
tff(fact_285_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),X_1)) ) ).

% linordered_field_no_ub
tff(fact_286_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X3)) ) ).

% linordered_field_no_lb
tff(fact_287_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_288_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_289_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_290_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_291_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_292_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_293_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_294_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_295_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_296_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_297_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_298_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_less_0_iff
tff(fact_299_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2))) ) ) ) ).

% divide_pos_pos
tff(fact_300_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),zero_zero(A))) ) ) ) ).

% divide_pos_neg
tff(fact_301_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),zero_zero(A))) ) ) ) ).

% divide_neg_pos
tff(fact_302_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2))) ) ) ) ).

% divide_neg_neg
tff(fact_303_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_304_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_305_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_306_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = M2 )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_307_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),M2)) ) ) ).

% div_less_dividend
tff(fact_308_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% less_divide_eq_1
tff(fact_309_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_310_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_311_div__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = zero_zero(nat) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = 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),M2),N)),N)) ) ) ) ).

% div_if
tff(fact_312_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_313_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ).

% compl_less_compl_iff
tff(fact_314_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_315_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_316_div__eq__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_317_div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = 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),M2),N)),N)) ) ) ) ).

% div_geq
tff(fact_318_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_319_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_320_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,uminus_uminus(A),X) = aa(A,A,uminus_uminus(A),Y2) )
        <=> ( X = Y2 ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_321_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),X)) = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_322_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_323_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_324_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),Y2))) ) ) ).

% compl_less_swap1
tff(fact_325_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),Y2)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y2)) ) ) ).

% compl_less_swap2
tff(fact_326_div__neg__pos__less0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),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_327_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_328_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int))) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_329_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : aa(option(A),nat,size_option(A,X),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_330_int__power__div__base,axiom,
    ! [M2: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),K),M2)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_331_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z)) ) ).

% one_less_nat_eq
tff(fact_332_Divides_Oadjust__mod__def,axiom,
    ! [R: int,L: int] :
      ( ( ( R = zero_zero(int) )
       => ( adjust_mod(L,R) = zero_zero(int) ) )
      & ( ( R != zero_zero(int) )
       => ( adjust_mod(L,R) = aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R) ) ) ) ).

% Divides.adjust_mod_def
tff(fact_333_le__div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = 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),M2),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_334_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B2) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)) ) )
        & ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(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_335_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,uminus_uminus(int),B2)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)) ) )
        & ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,uminus_uminus(int),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_336_zmod__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_337_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,N: nat,X: A,F2: fun(nat,A)] :
      ( ( ( N = zero_zero(nat) )
       => ( case_nat(A,X,F2,N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( case_nat(A,X,F2,N) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% Nitpick.case_nat_unfold
tff(fact_338_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_339_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_340_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),zero_zero(A)))
        <=> ( N = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_341_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% neg_le_iff_le
tff(fact_342_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) ) ) ).

% compl_le_compl_iff
tff(fact_343_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,A3) = zero_zero(A) ) ).

% mod_self
tff(fact_344_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,zero_zero(A)) = A3 ) ).

% mod_by_0
tff(fact_345_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ).

% mod_0
tff(fact_346_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_347_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(nat,nat,suc,M2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)) ) ).

% Suc_le_mono
tff(fact_348_le0,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% le0
tff(fact_349_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A3)) ).

% bot_nat_0.extremum
tff(fact_350_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_351_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_352_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_353_negative__zle,axiom,
    ! [N: nat,M2: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),aa(nat,int,semiring_1_of_nat(int),M2))) ).

% negative_zle
tff(fact_354_nat__int,axiom,
    ! [N: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),N)) = N ).

% nat_int
tff(fact_355_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% diff_ge_0_iff_ge
tff(fact_356_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% neg_0_le_iff_le
tff(fact_357_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% neg_le_0_iff_le
tff(fact_358_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% less_eq_neg_nonpos
tff(fact_359_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% neg_less_eq_nonneg
tff(fact_360_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_361_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_362_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_363_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_364_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% of_nat_le_iff
tff(fact_365_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_366_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% diff_is_0_eq
tff(fact_367_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_368_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa(int,nat,nat2,I) = zero_zero(nat) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int))) ) ).

% nat_0_iff
tff(fact_369_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
     => ( aa(int,nat,nat2,Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_370_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = Z ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = zero_zero(int) ) ) ) ).

% int_nat_eq
tff(fact_371_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_372_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_373_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ).

% zle_diff1_eq
tff(fact_374_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_375_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_376_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% zero_le_divide_1_iff
tff(fact_377_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% divide_le_0_1_iff
tff(fact_378_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M2)),zero_zero(A)))
        <=> ( M2 = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_379_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_380_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_381_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% zless_nat_conj
tff(fact_382_nat__zminus__int,axiom,
    ! [N: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_383_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_384_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_385_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

% zero_less_nat_eq
tff(fact_386_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).

% verit_comp_simplify1(2)
tff(fact_387_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% verit_la_disequality
tff(fact_388_le__refl,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).

% le_refl
tff(fact_389_le__trans,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K)) ) ) ).

% le_trans
tff(fact_390_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% eq_imp_le
tff(fact_391_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
       => ( M2 = N ) ) ) ).

% le_antisym
tff(fact_392_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
      | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)) ) ).

% nat_le_linear
tff(fact_393_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => ? [X2: nat] :
            ( pp(aa(nat,bool,P,X2))
            & ! [Y: nat] :
                ( pp(aa(nat,bool,P,Y))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X2)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_394_nat__mono,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Y2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y2))) ) ).

% nat_mono
tff(fact_395_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),N))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_le_iff
tff(fact_396_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),modulo_modulo(A,A3,B2)),A3)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_397_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M2,N)) = modulo_modulo(A,aa(nat,A,semiring_1_of_nat(A),M2),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mod
tff(fact_398_eq__nat__nat__iff,axiom,
    ! [Z: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z2) )
        <=> ( Z = Z2 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_399_all__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ! [X_13: nat] : pp(aa(nat,bool,P,X_13))
    <=> ! [X4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
         => pp(aa(nat,bool,P,aa(int,nat,nat2,X4))) ) ) ).

% all_nat
tff(fact_400_ex__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ? [X_13: nat] : pp(aa(nat,bool,P,X_13))
    <=> ? [X4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
          & pp(aa(nat,bool,P,aa(int,nat,nat2,X4))) ) ) ).

% ex_nat
tff(fact_401_zmod__le__nonneg__dividend,axiom,
    ! [M2: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,M2,K)),M2)) ) ).

% zmod_le_nonneg_dividend
tff(fact_402_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N3: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,F2,N3))) ) ) ) ).

% lift_Suc_mono_le
tff(fact_403_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N3: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N2))),aa(nat,A,F2,N2)))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N3)),aa(nat,A,F2,N))) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_404_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J2))) ) ) ).

% of_nat_mono
tff(fact_405_zle__int,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% zle_int
tff(fact_406_nat__int__comparison_I3_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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_407_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),W))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_408_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(int,nat,nat2,K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ) ) ).

% le_nat_iff
tff(fact_409_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A3,B2))) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_410_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( modulo_modulo(A,A3,B2) = A3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_411_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_412_int__eq__iff,axiom,
    ! [M2: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M2) = Z )
    <=> ( ( M2 = aa(int,nat,nat2,Z) )
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) ) ) ).

% int_eq_iff
tff(fact_413_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_414_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,K,L)),zero_zero(int))) ) ).

% neg_mod_sign
tff(fact_415_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( modulo_modulo(int,I,K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_416_pos__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A3,B2)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,A3,B2)),B2)) ) ) ).

% pos_mod_conj
tff(fact_417_neg__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,A3,B2)),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),modulo_modulo(int,A3,B2))) ) ) ).

% neg_mod_conj
tff(fact_418_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_419_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_420_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_421_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_422_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_423_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_424_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_425_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ).

% zero_le
tff(fact_426_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),zero_zero(A))) ) ).

% le_numeral_extra(3)
tff(fact_427_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B6: B,A7: B] :
          ( ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B6),A7))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A7),B6)) ) ) ).

% verit_comp_simplify1(3)
tff(fact_428_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),one_one(A))) ) ).

% le_numeral_extra(4)
tff(fact_429_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_430_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_431_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_432_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) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2)) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_433_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% le_minus_iff
tff(fact_434_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).

% minus_le_iff
tff(fact_435_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% le_imp_neg_le
tff(fact_436_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y2)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y2)) ) ) ).

% compl_le_swap2
tff(fact_437_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y2))) ) ) ).

% compl_le_swap1
tff(fact_438_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y2)),aa(A,A,uminus_uminus(A),X))) ) ) ).

% compl_mono
tff(fact_439_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% Suc_leD
tff(fact_440_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( M2 = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_441_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N))) ) ).

% le_SucI
tff(fact_442_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M6))
     => ? [M3: nat] : M6 = aa(nat,nat,suc,M3) ) ).

% Suc_le_D
tff(fact_443_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
        | ( M2 = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_444_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_445_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M2)) ) ).

% not_less_eq_eq
tff(fact_446_full__nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N2))
             => pp(aa(nat,bool,P,M)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% full_nat_induct
tff(fact_447_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( pp(aa(nat,bool,P,M2))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_448_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R2: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( ! [X2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X2),X2))
       => ( ! [X2: nat,Y3: nat,Z3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X2),Y3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,Y3),Z3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X2),Z3)) ) )
         => ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,N2),aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,M2),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_449_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_450_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
     => ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_451_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
    <=> ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_452_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% less_eq_nat.simps(1)
tff(fact_453_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% real_arch_simple
tff(fact_454_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J2: nat] :
      ( ! [I3: nat,J: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F2,I3)),aa(nat,nat,F2,J))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,F2,I)),aa(nat,nat,F2,J2))) ) ) ).

% less_mono_imp_le_mono
tff(fact_455_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( ( M2 != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% le_neq_implies_less
tff(fact_456_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | ( M2 = N ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% less_or_eq_imp_le
tff(fact_457_le__eq__less__or__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | ( M2 = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_458_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% less_imp_le_nat
tff(fact_459_nat__less__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
        & ( M2 != N ) ) ) ).

% nat_less_le
tff(fact_460_less__eq__int__code_I1_J,axiom,
    pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).

% less_eq_int_code(1)
tff(fact_461_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K) )
        <=> ( M2 = N ) ) ) ) ).

% eq_diff_iff
tff(fact_462_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).

% le_diff_iff
tff(fact_463_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_464_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),L))) ) ).

% diff_le_mono
tff(fact_465_diff__le__self,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),M2)) ).

% diff_le_self
tff(fact_466_le__diff__iff_H,axiom,
    ! [A3: nat,C2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),C2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),C2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),A3)) ) ) ) ).

% le_diff_iff'
tff(fact_467_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M2))) ) ).

% diff_le_mono2
tff(fact_468_div__le__dividend,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),M2)) ).

% div_le_dividend
tff(fact_469_div__le__mono,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),K))) ) ).

% div_le_mono
tff(fact_470_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),X))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),X)))) ).

% real_of_nat_div4
tff(fact_471_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : pp(aa(real,bool,aa(real,fun(real,bool),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),N)),aa(nat,real,semiring_1_of_nat(real),X))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),X)))),one_one(real))) ).

% real_of_nat_div3
tff(fact_472_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : pp(aa(real,bool,aa(real,fun(real,bool),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),N)),aa(nat,real,semiring_1_of_nat(real),X))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),X))))) ).

% real_of_nat_div2
tff(fact_473_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_474_nat__eq__iff,axiom,
    ! [W: int,M2: nat] :
      ( ( aa(int,nat,nat2,W) = M2 )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( W = aa(nat,int,semiring_1_of_nat(int),M2) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( M2 = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff
tff(fact_475_nat__eq__iff2,axiom,
    ! [M2: nat,W: int] :
      ( ( M2 = aa(int,nat,nat2,W) )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( W = aa(nat,int,semiring_1_of_nat(int),M2) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( M2 = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff2
tff(fact_476_nat__diff__distrib_H,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y2)) ) ) ) ).

% nat_diff_distrib'
tff(fact_477_nat__diff__distrib,axiom,
    ! [Z2: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),Z))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) ) ) ) ).

% nat_diff_distrib
tff(fact_478_nat__div__distrib_H,axiom,
    ! [Y2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y2)) ) ) ).

% nat_div_distrib'
tff(fact_479_nat__div__distrib,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y2)) ) ) ).

% nat_div_distrib
tff(fact_480_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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_481_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X23: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X23)) = aa(nat,A,F22,X23) ).

% old.nat.simps(5)
tff(fact_482_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_483_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,N: nat,M2: nat] :
          ( ( A3 != zero_zero(A) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) ) )
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ) ) ).

% power_diff_power_eq
tff(fact_484_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,B2)),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_485_nat__less__iff,axiom,
    ! [W: int,M2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W)),M2))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(nat,int,semiring_1_of_nat(int),M2))) ) ) ).

% nat_less_iff
tff(fact_486_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_487_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: nat,M2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
         => ( ( N != zero_zero(nat) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),N)))) ) ) ) ).

% inverse_of_nat_le
tff(fact_488_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_489_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_490_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_491_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,K,L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_492_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),modulo_modulo(int,K,L))) ) ).

% neg_mod_bound
tff(fact_493_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_494_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_495_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) ) ).

% not_one_le_zero
tff(fact_496_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_497_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one_class.zero_le_one
tff(fact_498_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_499_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_le_0_iff
tff(fact_500_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_501_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_502_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2))) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_503_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),zero_zero(A))) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_504_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),zero_zero(A))) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_505_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2))) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_506_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2))) ) ) ) ).

% divide_right_mono_neg
tff(fact_507_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% le_minus_one_simps(2)
tff(fact_508_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(4)
tff(fact_509_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% of_nat_0_le_iff
tff(fact_510_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N)) ) ).

% Suc_leI
tff(fact_511_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_le_eq
tff(fact_512_dec__induct,axiom,
    ! [I: nat,J2: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( pp(aa(nat,bool,P,I))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),J2))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) )
         => pp(aa(nat,bool,P,J2)) ) ) ) ).

% dec_induct
tff(fact_513_inc__induct,axiom,
    ! [I: nat,J2: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( pp(aa(nat,bool,P,J2))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),J2))
               => ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
                 => pp(aa(nat,bool,P,N2)) ) ) )
         => pp(aa(nat,bool,P,I)) ) ) ) ).

% inc_induct
tff(fact_514_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_le_lessD
tff(fact_515_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2)))
      <=> ( N = M2 ) ) ) ).

% le_less_Suc_eq
tff(fact_516_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% less_Suc_eq_le
tff(fact_517_less__eq__Suc__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M2)) ) ).

% less_eq_Suc_le
tff(fact_518_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N))) ) ).

% le_imp_less_Suc
tff(fact_519_ex__least__nat__le,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
            & ! [I4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K2))
               => ~ pp(aa(nat,bool,P,I4)) )
            & pp(aa(nat,bool,P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_520_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) ) ) ).

% Suc_diff_le
tff(fact_521_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ) ).

% less_diff_iff
tff(fact_522_diff__less__mono,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),A3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),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_523_Suc__div__le__mono,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M2)),N))) ).

% Suc_div_le_mono
tff(fact_524_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ~ ! [N2: nat] : K != aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% nonneg_int_cases
tff(fact_525_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ? [N2: nat] : K = aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% zero_le_imp_eq_int
tff(fact_526_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),K))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),K))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_le_induct
tff(fact_527_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% nat_mono_iff
tff(fact_528_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(int,nat,nat2,Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M2)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_529_int__minus,axiom,
    ! [N: nat,M2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)) = aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M2)))) ).

% int_minus
tff(fact_530_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),zero_zero(A))) ) ) ) ).

% divide_nonpos_pos
tff(fact_531_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2))) ) ) ) ).

% divide_nonpos_neg
tff(fact_532_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2))) ) ) ) ).

% divide_nonneg_pos
tff(fact_533_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),zero_zero(A))) ) ) ) ).

% divide_nonneg_neg
tff(fact_534_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% divide_le_cancel
tff(fact_535_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A,W: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y2),W))) ) ) ) ) ) ).

% frac_less2
tff(fact_536_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A,W: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y2),W))) ) ) ) ) ) ).

% frac_less
tff(fact_537_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y2: A,X: A,W: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y2),W))) ) ) ) ) ) ).

% frac_le
tff(fact_538_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) ) ) ) ).

% div_positive
tff(fact_539_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_540_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% le_minus_one_simps(1)
tff(fact_541_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(3)
tff(fact_542_zdiv__mono__strict,axiom,
    ! [A4: int,B5: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A4),B5))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => ( ( modulo_modulo(int,A4,N) = zero_zero(int) )
         => ( ( modulo_modulo(int,B5,N) = zero_zero(int) )
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_543_zmod__zminus1__eq__if,axiom,
    ! [A3: int,B2: int] :
      ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A3),B2) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A3),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),modulo_modulo(int,A3,B2)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_544_zmod__zminus2__eq__if,axiom,
    ! [A3: int,B2: int] :
      ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
       => ( modulo_modulo(int,A3,aa(int,int,uminus_uminus(int),B2)) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
       => ( modulo_modulo(int,A3,aa(int,int,uminus_uminus(int),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,A3,B2)),B2) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_545_ex__least__nat__less,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),N))
            & ! [I4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),K2))
               => ~ pp(aa(nat,bool,P,I4)) )
            & pp(aa(nat,bool,P,aa(nat,nat,suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_546_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).

% of_nat_diff
tff(fact_547_div__greater__zero__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% div_greater_zero_iff
tff(fact_548_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M2))) ) ) ).

% div_le_mono2
tff(fact_549_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_550_int__zle__neg,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M2))))
    <=> ( ( N = zero_zero(nat) )
        & ( M2 = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_551_negative__zle__0,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),zero_zero(int))) ).

% negative_zle_0
tff(fact_552_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ~ ! [N2: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% nonpos_int_cases
tff(fact_553_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
      <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_554_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_555_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int))) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_556_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_557_div__nonpos__pos__le0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),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_558_div__nonneg__neg__le0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),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_559_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))) ) ) ).

% div_positive_int
tff(fact_560_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ) ).

% div_int_pos_iff
tff(fact_561_zdiv__mono2__neg,axiom,
    ! [A3: int,B6: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B6),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),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_562_zdiv__mono1__neg,axiom,
    ! [A3: int,A7: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A7))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),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_563_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_564_zdiv__mono2,axiom,
    ! [A3: int,B6: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B6),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),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_565_zdiv__mono1,axiom,
    ! [A3: int,A7: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A7))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),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_566_split__nat,axiom,
    ! [P: fun(nat,bool),I: int] :
      ( pp(aa(nat,bool,P,aa(int,nat,nat2,I)))
    <=> ( ! [N5: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N5) )
           => pp(aa(nat,bool,P,N5)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
         => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ) ).

% split_nat
tff(fact_567_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% le_divide_eq_1
tff(fact_568_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_569_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))) ).

% not_zle_0_negative
tff(fact_570_verit__less__mono__div__int2,axiom,
    ! [A4: int,B5: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A4),B5))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),N)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_571_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),X)),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_572_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)) ) ) ) ) ).

% power_decreasing_iff
tff(fact_573_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_mono_iff
tff(fact_574_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y2)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y2)) ) ) ) ).

% power_increasing_iff
tff(fact_575_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_576_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W))) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_577_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),X)) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_578_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W))) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_579_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),X)) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_580_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% power_eq_0_iff
tff(fact_581_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y2)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y2)) ) ) ) ).

% power_strict_increasing_iff
tff(fact_582_ln__inj__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
       => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,ln_ln(real),Y2) )
        <=> ( X = Y2 ) ) ) ) ).

% ln_inj_iff
tff(fact_583_ln__less__cancel__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_584_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),N) = one_one(A) ) ).

% power_one
tff(fact_585_ln__le__cancel__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_586_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ).

% ln_less_zero_iff
tff(fact_587_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_588_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( aa(real,real,ln_ln(real),X) = zero_zero(real) )
      <=> ( X = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_589_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_590_mod__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( modulo_modulo(nat,M2,N) = M2 ) ) ).

% mod_less
tff(fact_591_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) )
          <=> ( M2 = N ) ) ) ) ).

% power_inject_exp
tff(fact_592_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,N)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_593_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_594_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_595_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ).

% ln_le_zero_iff
tff(fact_596_power__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_597_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),M2) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M2 = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_598_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_599_of__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M2),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),M2)),N) ) ).

% of_nat_power
tff(fact_600_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B2: nat,W: nat,X: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) = aa(nat,A,semiring_1_of_nat(A),X) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) = X ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_601_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: nat,B2: nat,W: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),X) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) )
        <=> ( X = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_602_mod__by__Suc__0,axiom,
    ! [M2: nat] : modulo_modulo(nat,M2,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_603_ln__diff__le,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y2)),Y2))) ) ) ).

% ln_diff_le
tff(fact_604_real__arch__pow,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N2))) ) ).

% real_arch_pow
tff(fact_605_real__arch__pow__inv,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N2)),Y2)) ) ) ).

% real_arch_pow_inv
tff(fact_606_ln__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X))) ) ).

% ln_ge_zero
tff(fact_607_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_608_ln__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),X)) ) ).

% ln_bound
tff(fact_609_verit__la__generic,axiom,
    ! [A3: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),X))
      | ( A3 = X )
      | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),A3)) ) ).

% verit_la_generic
tff(fact_610_complete__real,axiom,
    ! [S3: set(real)] :
      ( ? [X3: real] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),S3))
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),S3))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),Z4)) )
       => ? [Y3: real] :
            ( ! [X3: real] :
                ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),S3))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),Y3)) )
            & ! [Z4: real] :
                ( ! [X2: real] :
                    ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),S3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),Z4)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),Z4)) ) ) ) ) ).

% complete_real
tff(fact_611_less__eq__real__def,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
        | ( X = Y2 ) ) ) ).

% less_eq_real_def
tff(fact_612_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),X))),aa(real,real,uminus_uminus(real),X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_613_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),one_one(real)))) ) ).

% ln_le_minus_one
tff(fact_614_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,aa(real,fun(real,real),minus_minus(real),X),one_one(real)) )
       => ( X = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_615_ln__less__self,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),X)) ) ).

% ln_less_self
tff(fact_616_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_617_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),zero_zero(real))) ) ) ).

% ln_less_zero
tff(fact_618_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X))) ) ).

% ln_gt_zero
tff(fact_619_ln__div,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y2)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y2)) ) ) ) ).

% ln_div
tff(fact_620_mod__Suc__eq,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,modulo_modulo(nat,M2,N)),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% mod_Suc_eq
tff(fact_621_mod__Suc__Suc__eq,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,modulo_modulo(nat,M2,N))),N) = modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M2)),N) ).

% mod_Suc_Suc_eq
tff(fact_622_nat__power__less__imp__less,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_623_mod__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( aa(nat,nat,suc,modulo_modulo(nat,M2,N)) = N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M2),N) = zero_zero(nat) ) )
      & ( ( aa(nat,nat,suc,modulo_modulo(nat,M2,N)) != N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M2),N) = aa(nat,nat,suc,modulo_modulo(nat,M2,N)) ) ) ) ).

% mod_Suc
tff(fact_624_mod__induct,axiom,
    ! [P: fun(nat,bool),N: nat,P2: nat,M2: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),P2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),P2))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),P2))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,modulo_modulo(nat,aa(nat,nat,suc,N2),P2))) ) )
           => pp(aa(nat,bool,P,M2)) ) ) ) ) ).

% mod_induct
tff(fact_625_mod__less__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),modulo_modulo(nat,M2,N)),N)) ) ).

% mod_less_divisor
tff(fact_626_mod__Suc__le__divisor,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M2,aa(nat,nat,suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_627_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),K))) ) ).

% power_gt_expt
tff(fact_628_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( modulo_modulo(nat,M2,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N) ) ) ).

% mod_geq
tff(fact_629_mod__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => ( modulo_modulo(nat,M2,N) = M2 ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => ( modulo_modulo(nat,M2,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N) ) ) ) ).

% mod_if
tff(fact_630_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N))) ) ).

% nat_one_le_power
tff(fact_631_le__mod__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( modulo_modulo(nat,M2,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N) ) ) ).

% le_mod_geq
tff(fact_632_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_633_mod__le__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M2,N)),N)) ) ).

% mod_le_divisor
tff(fact_634_div__less__mono,axiom,
    ! [A4: nat,B5: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A4),B5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( modulo_modulo(nat,A4,N) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B5,N) = zero_zero(nat) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),N))) ) ) ) ) ).

% div_less_mono
tff(fact_635_nat__mod__distrib,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
       => ( aa(int,nat,nat2,modulo_modulo(int,X,Y2)) = modulo_modulo(nat,aa(int,nat,nat2,X),aa(int,nat,nat2,Y2)) ) ) ) ).

% nat_mod_distrib
tff(fact_636_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(int,nat,nat2,aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(int,nat,nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_637_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)) = 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),M2)),aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M2,N)))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% field_char_0_class.of_nat_div
tff(fact_638_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_639_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ).

% power_divide
tff(fact_640_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% zero_le_power
tff(fact_641_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono
tff(fact_642_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% zero_less_power
tff(fact_643_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% one_le_power
tff(fact_644_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_one_over
tff(fact_645_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_646_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% power_less_imp_less_base
tff(fact_647_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A))) ) ) ) ).

% power_le_one
tff(fact_648_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( A3 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_649_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% power_le_imp_le_base
tff(fact_650_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ) ).

% power_0_left
tff(fact_651_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)))) ) ) ).

% power_gt1
tff(fact_652_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ) ).

% power_less_imp_less_exp
tff(fact_653_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N6: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N6))) ) ) ) ).

% power_strict_increasing
tff(fact_654_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N6: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N6))) ) ) ) ).

% power_increasing
tff(fact_655_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_656_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),A3)) ) ) ) ).

% power_Suc_le_self
tff(fact_657_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).

% power_Suc_less_one
tff(fact_658_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N6: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N6)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).

% power_strict_decreasing
tff(fact_659_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N6: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N6)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).

% power_decreasing
tff(fact_660_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).

% power_le_imp_le_exp
tff(fact_661_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
              <=> ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_662_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
               => ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_663_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).

% self_le_power
tff(fact_664_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).

% one_less_power
tff(fact_665_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,N: nat,M2: nat] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% power_diff
tff(fact_666_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ) ).

% power_strict_mono
tff(fact_667_zdiff__int__split,axiom,
    ! [P: fun(int,bool),X: nat,Y2: nat] :
      ( pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Y2))))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y2),X))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y2)))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y2))
         => pp(aa(int,bool,P,zero_zero(int))) ) ) ) ).

% zdiff_int_split
tff(fact_668_Bolzano,axiom,
    ! [A3: real,B2: real,P: fun(real,fun(real,bool))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
     => ( ! [A6: real,B4: real,C3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),P,A6),B4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),P,B4),C3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A6),B4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B4),C3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),P,A6),C3)) ) ) ) )
       => ( ! [X2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2))
               => ? [D3: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
                    & ! [A6: real,B4: real] :
                        ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A6),X2))
                          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B4))
                          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B4),A6)),D3)) )
                       => pp(aa(real,bool,aa(real,fun(real,bool),P,A6),B4)) ) ) ) )
         => pp(aa(real,bool,aa(real,fun(real,bool),P,A3),B2)) ) ) ) ).

% Bolzano
tff(fact_669_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M2: nat,N: nat] :
      ( ! [M3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M3),zero_zero(nat)))
     => ( ! [M3: nat,N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N2),modulo_modulo(nat,M3,N2)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M3),N2)) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M2),N)) ) ) ).

% gcd_nat_induct
tff(fact_670_realpow__pos__nth,axiom,
    ! [N: nat,A3: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => ? [R3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),N) = A3 ) ) ) ) ).

% realpow_pos_nth
tff(fact_671_realpow__pos__nth__unique,axiom,
    ! [N: nat,A3: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => ? [X2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X2))
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X2),N) = A3 )
            & ! [Y: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N) = A3 ) )
               => ( Y = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_672_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P,case_nat(A,F1,F22,Nat)))
    <=> ( ( ( Nat = zero_zero(nat) )
         => pp(aa(A,bool,P,F1)) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => pp(aa(A,bool,P,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(1)
tff(fact_673_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P,case_nat(A,F1,F22,Nat)))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ pp(aa(A,bool,P,F1)) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ pp(aa(A,bool,P,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(2)
tff(fact_674_imp__le__cong,axiom,
    ! [X: int,X5: int,P: bool,P3: bool] :
      ( ( X = X5 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
           => pp(P) )
        <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
           => pp(P3) ) ) ) ) ).

% imp_le_cong
tff(fact_675_conj__le__cong,axiom,
    ! [X: int,X5: int,P: bool,P3: bool] :
      ( ( X = X5 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
            & pp(P) )
        <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
            & pp(P3) ) ) ) ) ).

% conj_le_cong
tff(fact_676_arsinh__minus__real,axiom,
    ! [X: real] : aa(real,real,arsinh(real),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,arsinh(real),X)) ).

% arsinh_minus_real
tff(fact_677_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [Z4: A] :
            ! [X2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X2))
             => ( pp(aa(A,bool,P,X2))
              <=> pp(aa(A,bool,P3,X2)) ) )
         => ( ? [Z4: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X2))
               => ( pp(aa(A,bool,Q,X2))
                <=> pp(aa(A,bool,Q2,X2)) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
               => ( ( pp(aa(A,bool,P,X3))
                    & pp(aa(A,bool,Q,X3)) )
                <=> ( pp(aa(A,bool,P3,X3))
                    & pp(aa(A,bool,Q2,X3)) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_678_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [Z4: A] :
            ! [X2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X2))
             => ( pp(aa(A,bool,P,X2))
              <=> pp(aa(A,bool,P3,X2)) ) )
         => ( ? [Z4: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X2))
               => ( pp(aa(A,bool,Q,X2))
                <=> pp(aa(A,bool,Q2,X2)) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
               => ( ( pp(aa(A,bool,P,X3))
                    | pp(aa(A,bool,Q,X3)) )
                <=> ( pp(aa(A,bool,P3,X3))
                    | pp(aa(A,bool,Q2,X3)) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_679_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
         => ( X3 != T2 ) ) ) ).

% pinf(3)
tff(fact_680_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
         => ( X3 != T2 ) ) ) ).

% pinf(4)
tff(fact_681_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),T2)) ) ) ).

% pinf(5)
tff(fact_682_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),X3)) ) ) ).

% pinf(7)
tff(fact_683_pinf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F4: D] :
        ? [Z3: C] :
        ! [X3: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),Z3),X3))
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_684_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [Z4: A] :
            ! [X2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z4))
             => ( pp(aa(A,bool,P,X2))
              <=> pp(aa(A,bool,P3,X2)) ) )
         => ( ? [Z4: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z4))
               => ( pp(aa(A,bool,Q,X2))
                <=> pp(aa(A,bool,Q2,X2)) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
               => ( ( pp(aa(A,bool,P,X3))
                    & pp(aa(A,bool,Q,X3)) )
                <=> ( pp(aa(A,bool,P3,X3))
                    & pp(aa(A,bool,Q2,X3)) ) ) ) ) ) ) ).

% minf(1)
tff(fact_685_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [Z4: A] :
            ! [X2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z4))
             => ( pp(aa(A,bool,P,X2))
              <=> pp(aa(A,bool,P3,X2)) ) )
         => ( ? [Z4: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z4))
               => ( pp(aa(A,bool,Q,X2))
                <=> pp(aa(A,bool,Q2,X2)) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
               => ( ( pp(aa(A,bool,P,X3))
                    | pp(aa(A,bool,Q,X3)) )
                <=> ( pp(aa(A,bool,P3,X3))
                    | pp(aa(A,bool,Q2,X3)) ) ) ) ) ) ) ).

% minf(2)
tff(fact_686_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
         => ( X3 != T2 ) ) ) ).

% minf(3)
tff(fact_687_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
         => ( X3 != T2 ) ) ) ).

% minf(4)
tff(fact_688_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),T2)) ) ) ).

% minf(5)
tff(fact_689_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),X3)) ) ) ).

% minf(7)
tff(fact_690_minf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F4: D] :
        ? [Z3: C] :
        ! [X3: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),X3),Z3))
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_691_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),T2)) ) ) ).

% pinf(6)
tff(fact_692_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X3)) ) ) ).

% pinf(8)
tff(fact_693_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),T2)) ) ) ).

% minf(6)
tff(fact_694_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X3)) ) ) ).

% minf(8)
tff(fact_695_realpow__pos__nth2,axiom,
    ! [A3: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ? [R3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),aa(nat,nat,suc,N)) = A3 ) ) ) ).

% realpow_pos_nth2
tff(fact_696_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(nat,real,semiring_1_of_nat(real),N)))),N)),aa(real,real,exp(real),aa(real,real,uminus_uminus(real),X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_697_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F2: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,Y3)),B2)) )
       => ? [X2: A] :
            ( pp(aa(A,bool,P,X2))
            & ! [Y: A] :
                ( pp(aa(A,bool,P,Y))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,Y)),aa(A,nat,F2,X2))) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_698_nat__descend__induct,axiom,
    ! [N: nat,P: fun(nat,bool),M2: nat] :
      ( ! [K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
         => pp(aa(nat,bool,P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
           => ( ! [I4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),I4))
                 => pp(aa(nat,bool,P,I4)) )
             => pp(aa(nat,bool,P,K2)) ) )
       => pp(aa(nat,bool,P,M2)) ) ) ).

% nat_descend_induct
tff(fact_699_nat__ivt__aux,axiom,
    ! [N: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F2,N)))
         => ? [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_700_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
       => ( aa(real,real,ln_ln(real),aa(real,real,root(N),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% ln_root
tff(fact_701_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
            | ( X = Y2 ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_702_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ).

% linorder_le_less_linear
tff(fact_703_order__less__le__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,B2)),C2))
           => ( ! [X2: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,X2)),aa(A,C,F2,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,A3)),C2)) ) ) ) ) ).

% order_less_le_subst2
tff(fact_704_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X2: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X2)),aa(B,A,F2,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F2,C2))) ) ) ) ) ).

% order_less_le_subst1
tff(fact_705_order__le__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,B2)),C2))
           => ( ! [X2: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,X2)),aa(A,C,F2,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,A3)),C2)) ) ) ) ) ).

% order_le_less_subst2
tff(fact_706_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_707_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_708_exp__inj__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa(real,real,exp(real),X) = aa(real,real,exp(real),Y2) )
    <=> ( X = Y2 ) ) ).

% exp_inj_iff
tff(fact_709_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_710_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_711_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_712_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_713_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_714_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_715_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_716_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_717_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% abs_of_nat
tff(fact_718_exp__less__cancel__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,exp(real),X)),aa(real,real,exp(real),Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ).

% exp_less_cancel_iff
tff(fact_719_exp__less__mono,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,exp(real),X)),aa(real,real,exp(real),Y2))) ) ).

% exp_less_mono
tff(fact_720_exp__le__cancel__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,exp(real),X)),aa(real,real,exp(real),Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ).

% exp_le_cancel_iff
tff(fact_721_ln__exp,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),aa(real,real,exp(real),X)) = X ).

% ln_exp
tff(fact_722_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_723_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% abs_le_self_iff
tff(fact_724_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_725_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_726_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_727_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)) = aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% abs_power_minus
tff(fact_728_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,exp(A),zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_729_real__root__Suc__0,axiom,
    ! [X: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),X) = X ).

% real_root_Suc_0
tff(fact_730_root__0,axiom,
    ! [X: real] : aa(real,real,root(zero_zero(nat)),X) = zero_zero(real) ).

% root_0
tff(fact_731_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = aa(real,real,root(N),Y2) )
      <=> ( X = Y2 ) ) ) ).

% real_root_eq_iff
tff(fact_732_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,exp(real),X) = one_one(real) )
    <=> ( X = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_733_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_734_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_735_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_736_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_737_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ) ).

% real_root_less_iff
tff(fact_738_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ) ).

% real_root_le_iff
tff(fact_739_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_740_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = one_one(real) )
      <=> ( X = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_741_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,exp(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% one_less_exp_iff
tff(fact_742_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,exp(real),X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% exp_less_one_iff
tff(fact_743_exp__le__one__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,exp(real),X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% exp_le_one_iff
tff(fact_744_one__le__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,exp(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% one_le_exp_iff
tff(fact_745_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int)))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_746_exp__ln,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,exp(real),aa(real,real,ln_ln(real),X)) = X ) ) ).

% exp_ln
tff(fact_747_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,exp(real),aa(real,real,ln_ln(real),X)) = X )
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% exp_ln_iff
tff(fact_748_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N)))
        <=> ( ( A3 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_749_real__root__gt__0__iff,axiom,
    ! [N: nat,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,root(N),Y2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2)) ) ) ).

% real_root_gt_0_iff
tff(fact_750_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ) ).

% real_root_lt_0_iff
tff(fact_751_real__root__ge__0__iff,axiom,
    ! [N: nat,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,root(N),Y2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2)) ) ) ).

% real_root_ge_0_iff
tff(fact_752_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ) ).

% real_root_le_0_iff
tff(fact_753_real__root__gt__1__iff,axiom,
    ! [N: nat,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,root(N),Y2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),Y2)) ) ) ).

% real_root_gt_1_iff
tff(fact_754_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ).

% real_root_lt_1_iff
tff(fact_755_real__root__ge__1__iff,axiom,
    ! [N: nat,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,root(N),Y2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y2)) ) ) ).

% real_root_ge_1_iff
tff(fact_756_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ).

% real_root_le_1_iff
tff(fact_757_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_758_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_self
tff(fact_759_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% abs_le_D1
tff(fact_760_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_761_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_762_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_763_exp__less__cancel,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,exp(real),X)),aa(real,real,exp(real),Y2)))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ).

% exp_less_cancel
tff(fact_764_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,abs_abs(A),X) = aa(A,A,abs_abs(A),Y2) )
        <=> ( ( X = Y2 )
            | ( X = aa(A,A,uminus_uminus(A),Y2) ) ) ) ) ).

% abs_eq_iff
tff(fact_765_real__root__divide,axiom,
    ! [N: nat,X: real,Y2: real] : aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y2)) ).

% real_root_divide
tff(fact_766_real__root__minus,axiom,
    ! [N: nat,X: real] : aa(real,real,root(N),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,root(N),X)) ).

% real_root_minus
tff(fact_767_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_768_ln__unique,axiom,
    ! [Y2: real,X: real] :
      ( ( aa(real,real,exp(real),Y2) = X )
     => ( aa(real,real,ln_ln(real),X) = Y2 ) ) ).

% ln_unique
tff(fact_769_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_zero
tff(fact_770_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_771_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))) ) ).

% abs_not_less_zero
tff(fact_772_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_773_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_774_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_775_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_776_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_minus_self
tff(fact_777_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).

% abs_le_iff
tff(fact_778_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ).

% abs_le_D2
tff(fact_779_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)) ) ) ) ).

% abs_leI
tff(fact_780_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).

% abs_less_iff
tff(fact_781_not__exp__less__zero,axiom,
    ! [X: real] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,exp(real),X)),zero_zero(real))) ).

% not_exp_less_zero
tff(fact_782_exp__gt__zero,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,exp(real),X))) ).

% exp_gt_zero
tff(fact_783_exp__total,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
     => ? [X2: real] : aa(real,real,exp(real),X2) = Y2 ) ).

% exp_total
tff(fact_784_exp__ge__zero,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,exp(real),X))) ).

% exp_ge_zero
tff(fact_785_not__exp__le__zero,axiom,
    ! [X: real] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,exp(real),X)),zero_zero(real))) ).

% not_exp_le_zero
tff(fact_786_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y2)) ) ).

% exp_diff
tff(fact_787_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E)) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_788_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_789_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,abs_abs(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            & ( ( B2 = A3 )
              | ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_790_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A3) = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
            & ( ( A3 = B2 )
              | ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_791_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N))) ) ).

% zero_le_power_abs
tff(fact_792_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),X)),Y2) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)) ) ) ) ).

% abs_div_pos
tff(fact_793_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X3) = aa(A,A,uminus_uminus(A),X3) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X3) = X3 ) ) ) ) ).

% abs_if_raw
tff(fact_794_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_795_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ) ).

% abs_if
tff(fact_796_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y2))) ) ) ).

% real_root_less_mono
tff(fact_797_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y2))) ) ) ).

% real_root_le_mono
tff(fact_798_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),K) ) ) ).

% real_root_power
tff(fact_799_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,exp(real),X))) ) ).

% exp_gt_one
tff(fact_800_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I) = aa(int,int,uminus_uminus(int),I) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I) = I ) ) ) ).

% zabs_def
tff(fact_801_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),modulo_modulo(int,K,L))),aa(int,int,abs_abs(int),L))) ) ).

% abs_mod_less
tff(fact_802_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X)) ) ).

% lt_ex
tff(fact_803_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X_1)) ) ).

% gt_ex
tff(fact_804_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ? [Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),Y2)) ) ) ) ).

% dense
tff(fact_805_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( X != Y2 ) ) ) ).

% less_imp_neq
tff(fact_806_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% order.asym
tff(fact_807_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% ord_eq_less_trans
tff(fact_808_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% ord_less_eq_trans
tff(fact_809_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A3: A] :
          ( ! [X2: A] :
              ( ! [Y: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X2))
                 => pp(aa(A,bool,P,Y)) )
             => pp(aa(A,bool,P,X2)) )
         => pp(aa(A,bool,P,A3)) ) ) ).

% less_induct
tff(fact_810_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y2: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
          <=> ( X = Y2 ) ) ) ) ).

% antisym_conv3
tff(fact_811_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( ( X != Y2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ) ).

% linorder_cases
tff(fact_812_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% dual_order.asym
tff(fact_813_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).

% dual_order.irrefl
tff(fact_814_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
        <=> ? [N5: A] :
              ( pp(aa(A,bool,P,N5))
              & ! [M5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M5),N5))
                 => ~ pp(aa(A,bool,P,M5)) ) ) ) ) ).

% exists_least_iff
tff(fact_815_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A3: A,B2: A] :
          ( ! [A6: A,B4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A6),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B4)) )
         => ( ! [A6: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,A6),A6))
           => ( ! [A6: A,B4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),P,B4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B4)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A3),B2)) ) ) ) ) ).

% linorder_less_wlog
tff(fact_816_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% order.strict_trans
tff(fact_817_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
            | ( X = Y2 ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_818_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% dual_order.strict_trans
tff(fact_819_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( A3 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_820_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( A3 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_821_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A] :
          ( ( X != Y2 )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ) ).

% linorder_neqE
tff(fact_822_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ).

% order_less_asym
tff(fact_823_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A] :
          ( ( X != Y2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ) ).

% linorder_neq_iff
tff(fact_824_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% order_less_asym'
tff(fact_825_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_less_trans
tff(fact_826_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) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X2: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X2)),aa(B,A,F2,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F2,C2))) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_827_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F2: fun(A,B),C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X2: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,A3)),C2)) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_828_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).

% order_less_irrefl
tff(fact_829_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X2: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X2)),aa(B,A,F2,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F2,C2))) ) ) ) ) ).

% order_less_subst1
tff(fact_830_order__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,B2)),C2))
           => ( ! [X2: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,X2)),aa(A,C,F2,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,A3)),C2)) ) ) ) ) ).

% order_less_subst2
tff(fact_831_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ).

% order_less_not_sym
tff(fact_832_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y2: A,P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
           => pp(P) ) ) ) ).

% order_less_imp_triv
tff(fact_833_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
          | ( X = Y2 )
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ).

% linorder_less_linear
tff(fact_834_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( X != Y2 ) ) ) ).

% order_less_imp_not_eq
tff(fact_835_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( Y2 != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_836_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ).

% order_less_imp_not_less
tff(fact_837_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_gt_zero
tff(fact_838_real__root__strict__decreasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N6),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_839_lemma__exp__total,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y2))
     => ? [X2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X2))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),aa(real,real,aa(real,fun(real,real),minus_minus(real),Y2),one_one(real))))
          & ( aa(real,real,exp(real),X2) = Y2 ) ) ) ).

% lemma_exp_total
tff(fact_840_ln__ge__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,exp(real),Y2)),X)) ) ) ).

% ln_ge_iff
tff(fact_841_ln__x__over__x__mono,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,exp(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Y2)),Y2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),X))) ) ) ).

% ln_x_over_x_mono
tff(fact_842_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_843_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_844_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_pos_pos
tff(fact_845_real__root__strict__increasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N6),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_846_real__root__decreasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N6),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_847_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_848_real__root__pos__unique,axiom,
    ! [N: nat,Y2: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),N) = X )
         => ( aa(real,real,root(N),X) = Y2 ) ) ) ) ).

% real_root_pos_unique
tff(fact_849_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_850_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(nat,A,semiring_1_of_nat(A),N)))),N) = aa(A,A,exp(A),X) ) ) ) ).

% exp_divide_power_eq
tff(fact_851_nat__abs__int__diff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
       => ( 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)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A3) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
       => ( 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)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_852_real__root__increasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N6),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_853_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),I3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N)) )
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,M2)),K))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F2,N)))
           => ? [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),I3))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
                & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_854_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2)) ) ) ).

% leD
tff(fact_855_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) ) ) ).

% leI
tff(fact_856_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            | ( A3 = B2 ) ) ) ) ).

% nless_le
tff(fact_857_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
          <=> ( X = Y2 ) ) ) ) ).

% antisym_conv1
tff(fact_858_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
          <=> ( X = Y2 ) ) ) ) ).

% antisym_conv2
tff(fact_859_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y2: A] :
          ( ! [X2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X2)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Z)) ) ) ).

% dense_ge
tff(fact_860_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y2: A,Z: A] :
          ( ! [X2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Z)) ) ) ).

% dense_le
tff(fact_861_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) ) ) ) ).

% less_le_not_le
tff(fact_862_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y2: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2)) ) ) ).

% not_le_imp_less
tff(fact_863_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
            | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_864_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_865_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% order.strict_trans1
tff(fact_866_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% order.strict_trans2
tff(fact_867_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% order.strict_iff_not
tff(fact_868_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X))
         => ( ! [W2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),W2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W2),X))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),W2)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Z)) ) ) ) ).

% dense_ge_bounded
tff(fact_869_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( ! [W2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),W2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W2),Y2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Z)) ) ) ) ).

% dense_le_bounded
tff(fact_870_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
            | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_871_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_872_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% dual_order.strict_trans1
tff(fact_873_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% dual_order.strict_trans2
tff(fact_874_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_875_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% order.strict_implies_order
tff(fact_876_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% dual_order.strict_implies_order
tff(fact_877_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
            | ( X = Y2 ) ) ) ) ).

% order_le_less
tff(fact_878_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
            & ( X != Y2 ) ) ) ) ).

% order_less_le
tff(fact_879_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ) ).

% linorder_not_le
tff(fact_880_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) ) ) ).

% linorder_not_less
tff(fact_881_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) ) ) ).

% order_less_imp_le
tff(fact_882_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( A3 != B2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% order_le_neq_trans
tff(fact_883_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% order_neq_le_trans
tff(fact_884_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_le_less_trans
tff(fact_885_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_less_le_trans
tff(fact_886_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X2: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X2)),aa(B,A,F2,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F2,C2))) ) ) ) ) ).

% order_le_less_subst1
tff(fact_887_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(nat,real,semiring_1_of_nat(real),N)))),N)),aa(real,real,exp(real),X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_888_nat0__intermed__int__val,axiom,
    ! [N: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F2,I3)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F2,N)))
         => ? [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_889_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( aa(real,real,root(N),X) = aa(real,real,aa(real,fun(real,real),powr(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% root_powr_inverse
tff(fact_890_log__root,axiom,
    ! [N: nat,A3: real,B2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => ( aa(real,real,log2(B2),aa(real,real,root(N),A3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log2(B2),A3)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% log_root
tff(fact_891_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,P,A3))
           => ( ~ pp(aa(A,bool,P,B2))
             => ? [C3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
                  & ! [X3: A] :
                      ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),C3)) )
                     => pp(aa(A,bool,P,X3)) )
                  & ! [D3: A] :
                      ( ! [X2: A] :
                          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X2))
                            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),D3)) )
                         => pp(aa(A,bool,P,X2)) )
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3)) ) ) ) ) ) ) ).

% complete_interval
tff(fact_892_log__of__power__le,axiom,
    ! [M2: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log2(B2),aa(nat,real,semiring_1_of_nat(real),M2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_le
tff(fact_893_verit__le__mono__div__int,axiom,
    ! [A4: int,B5: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A4),B5))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,B5,N)),zero_zero(int)),one_one(int),zero_zero(int)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_894_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_895_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_896_verit__le__mono__div,axiom,
    ! [A4: nat,B5: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A4),B5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),modulo_modulo(nat,B5,N)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),N))) ) ) ).

% verit_le_mono_div
tff(fact_897_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2)
     => ~ ! [N2: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) ) ).

% even_odd_cases
tff(fact_898_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_899_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_900_abs__exp__cancel,axiom,
    ! [X: real] : aa(real,real,abs_abs(real),aa(real,real,exp(real),X)) = aa(real,real,exp(real),X) ).

% abs_exp_cancel
tff(fact_901_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_cancel_left
tff(fact_902_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_cancel_right
tff(fact_903_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_904_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_905_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2) )
        <=> ( ( X = zero_zero(A) )
            & ( Y2 = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_906_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y2 = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_907_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_908_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_909_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_910_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_911_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_912_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_913_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_cancel_right
tff(fact_914_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_cancel_left
tff(fact_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_add__Suc__right,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) ).

% add_Suc_right
tff(fact_926_Nat_Oadd__0__right,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),zero_zero(nat)) = M2 ).

% Nat.add_0_right
tff(fact_927_add__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = zero_zero(nat) )
    <=> ( ( M2 = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_928_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% nat_add_left_cancel_less
tff(fact_929_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% nat_add_left_cancel_le
tff(fact_930_diff__diff__left,axiom,
    ! [I: nat,J2: 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),J2)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K)) ).

% diff_diff_left
tff(fact_931_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W: A,Z: A] :
          ( ( aa(A,A,aa(A,fun(A,A),powr(A),W),Z) = zero_zero(A) )
        <=> ( W = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_932_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),powr(A),zero_zero(A)),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_933_powr__one__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),powr(A),one_one(A)),A3) = one_one(A) ) ).

% powr_one_eq_one
tff(fact_934_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% add_le_same_cancel1
tff(fact_935_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% add_le_same_cancel2
tff(fact_936_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel1
tff(fact_937_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel2
tff(fact_938_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_939_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_940_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_941_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_942_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel2
tff(fact_943_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel1
tff(fact_944_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% add_less_same_cancel2
tff(fact_945_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% add_less_same_cancel1
tff(fact_946_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_947_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_948_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_949_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_950_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_951_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_952_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_953_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_add
tff(fact_954_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% add_gr_0
tff(fact_955_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),powr(A),X),zero_zero(A)) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),powr(A),X),zero_zero(A)) = one_one(A) ) ) ) ) ).

% powr_zero_eq_one
tff(fact_956_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J2)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_957_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K)),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_958_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),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)),J2) ) ) ).

% Nat.diff_diff_right
tff(fact_959_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),X)) ) ) ).

% artanh_minus_real
tff(fact_960_powr__gt__zero,axiom,
    ! [X: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)))
    <=> ( X != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_961_real__add__minus__iff,axiom,
    ! [X: real,A3: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),A3)) = zero_zero(real) )
    <=> ( X = A3 ) ) ).

% real_add_minus_iff
tff(fact_962_powr__nonneg__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),A3),X)),zero_zero(real)))
    <=> ( A3 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_963_powr__less__cancel__iff,axiom,
    ! [X: real,A3: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),X),B2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2)) ) ) ).

% powr_less_cancel_iff
tff(fact_964_log__one,axiom,
    ! [A3: real] : aa(real,real,log2(A3),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_965_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_966_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_967_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M2: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M2)) ) ).

% of_nat_Suc
tff(fact_968_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J2))
     => ( 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),J2),K))),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_969_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J2))
     => ( 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),J2),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,J2)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_970_powr__eq__one__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( ( aa(real,real,aa(real,fun(real,real),powr(real),A3),X) = one_one(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_971_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,aa(real,fun(real,real),powr(real),X),one_one(real)) = X )
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% powr_one_gt_zero_iff
tff(fact_972_powr__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),powr(real),X),one_one(real)) = X ) ) ).

% powr_one
tff(fact_973_powr__le__cancel__iff,axiom,
    ! [X: real,A3: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),X),B2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2)) ) ) ).

% powr_le_cancel_iff
tff(fact_974_log__eq__one,axiom,
    ! [A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log2(A3),A3) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_975_log__less__cancel__iff,axiom,
    ! [A3: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log2(A3),X)),aa(real,real,log2(A3),Y2)))
          <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_976_log__less__one__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log2(A3),X)),one_one(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),A3)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_977_one__less__log__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,log2(A3),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_978_log__less__zero__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log2(A3),X)),zero_zero(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_979_zero__less__log__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,log2(A3),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_980_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ).

% zle_add1_eq_le
tff(fact_981_zero__le__log__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,log2(A3),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_982_log__le__zero__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log2(A3),X)),zero_zero(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_983_one__le__log__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,log2(A3),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_984_log__le__one__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log2(A3),X)),one_one(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),A3)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_985_log__le__cancel__iff,axiom,
    ! [A3: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log2(A3),X)),aa(real,real,log2(A3),Y2)))
          <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_986_log__powr__cancel,axiom,
    ! [A3: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log2(A3),aa(real,real,aa(real,fun(real,real),powr(real),A3),Y2)) = Y2 ) ) ) ).

% log_powr_cancel
tff(fact_987_powr__log__cancel,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),powr(real),A3),aa(real,real,log2(A3),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_988_log__pow__cancel,axiom,
    ! [A3: real,B2: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log2(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_989_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_990_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_991_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J2: A,K: A,L: A] :
          ( ( ( I = J2 )
            & ( 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),J2),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_992_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_993_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B5: A,K: A,B2: A,A3: A] :
          ( ( B5 = 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),B5) = 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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_powr__powr__swap,axiom,
    ! [X: real,A3: real,B2: real] : aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),B2) = aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,aa(real,fun(real,real),powr(real),X),B2)),A3) ).

% powr_powr_swap
tff(fact_1002_zadd__int__left,axiom,
    ! [M2: nat,N: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),Z)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))),Z) ).

% zadd_int_left
tff(fact_1003_int__plus,axiom,
    ! [N: nat,M2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M2)) ).

% int_plus
tff(fact_1004_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_1005_log__base__powr,axiom,
    ! [A3: real,B2: real,X: real] :
      ( ( A3 != zero_zero(real) )
     => ( aa(real,real,log2(aa(real,real,aa(real,fun(real,real),powr(real),A3),B2)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log2(A3),X)),B2) ) ) ).

% log_base_powr
tff(fact_1006_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_1007_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J2: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J2))
            & ( K = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1008_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J2: A,K: A,L: A] :
          ( ( ( I = J2 )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1009_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J2: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1010_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1011_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1012_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1013_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1014_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1015_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_imp_le_left
tff(fact_1016_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_imp_le_right
tff(fact_1017_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_1018_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_1019_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_1020_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_1021_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_imp_less_right
tff(fact_1022_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_imp_less_left
tff(fact_1023_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1024_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1025_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1026_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J2: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J2))
            & ( K = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1027_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J2: A,K: A,L: A] :
          ( ( ( I = J2 )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1028_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J2: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_add__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) ).

% add_Suc
tff(fact_1045_add__Suc__shift,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_1046_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),A3: nat,B2: nat] :
      ( ! [A6: nat,B4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A6),B4))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,B4),A6)) )
     => ( ! [A6: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A6),zero_zero(nat)))
       => ( ! [A6: nat,B4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A6),B4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A6),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A6),B4))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A3),B2)) ) ) ) ).

% Euclid_induct
tff(fact_1047_add__eq__self__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = M2 )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_1048_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).

% plus_nat.add_0
tff(fact_1049_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% less_add_eq_less
tff(fact_1050_trans__less__add2,axiom,
    ! [I: nat,J2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),J2))) ) ).

% trans_less_add2
tff(fact_1051_trans__less__add1,axiom,
    ! [I: nat,J2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),M2))) ) ).

% trans_less_add1
tff(fact_1052_add__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K))) ) ).

% add_less_mono1
tff(fact_1053_not__add__less2,axiom,
    ! [J2: nat,I: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),I)),I)) ).

% not_add_less2
tff(fact_1054_not__add__less1,axiom,
    ! [I: nat,J2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J2)),I)) ).

% not_add_less1
tff(fact_1055_add__less__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),L))) ) ) ).

% add_less_mono
tff(fact_1056_add__lessD1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J2)),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K)) ) ).

% add_lessD1
tff(fact_1057_plus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K),zero_zero(int)) = K ).

% plus_int_code(1)
tff(fact_1058_plus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),L) = L ).

% plus_int_code(2)
tff(fact_1059_add__leE,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K)),N))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).

% add_leE
tff(fact_1060_le__add1,axiom,
    ! [N: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))) ).

% le_add1
tff(fact_1061_le__add2,axiom,
    ! [N: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) ).

% le_add2
tff(fact_1062_add__leD1,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% add_leD1
tff(fact_1063_add__leD2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).

% add_leD2
tff(fact_1064_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
     => ? [N2: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2) ) ).

% le_Suc_ex
tff(fact_1065_add__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),L))) ) ) ).

% add_le_mono
tff(fact_1066_add__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K))) ) ).

% add_le_mono1
tff(fact_1067_trans__le__add1,axiom,
    ! [I: nat,J2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),M2))) ) ).

% trans_le_add1
tff(fact_1068_trans__le__add2,axiom,
    ! [I: nat,J2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),J2))) ) ).

% trans_le_add2
tff(fact_1069_nat__le__iff__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
    <=> ? [K3: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K3) ) ).

% nat_le_iff_add
tff(fact_1070_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ).

% Nat.diff_cancel
tff(fact_1071_diff__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ).

% diff_cancel2
tff(fact_1072_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),N) = M2 ).

% diff_add_inverse
tff(fact_1073_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),N) = M2 ).

% diff_add_inverse2
tff(fact_1074_less__log__iff,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(real,real,log2(B2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2)),X)) ) ) ) ).

% less_log_iff
tff(fact_1075_log__less__iff,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log2(B2),X)),Y2))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2))) ) ) ) ).

% log_less_iff
tff(fact_1076_less__powr__iff,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log2(B2),X)),Y2)) ) ) ) ).

% less_powr_iff
tff(fact_1077_powr__less__iff,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2)),X))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(real,real,log2(B2),X))) ) ) ) ).

% powr_less_iff
tff(fact_1078_nat__add__distrib,axiom,
    ! [Z: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) ) ) ) ).

% nat_add_distrib
tff(fact_1079_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_1080_powr__le__iff,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2)),X))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),aa(real,real,log2(B2),X))) ) ) ) ).

% powr_le_iff
tff(fact_1081_le__powr__iff,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log2(B2),X)),Y2)) ) ) ) ).

% le_powr_iff
tff(fact_1082_log__le__iff,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log2(B2),X)),Y2))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2))) ) ) ) ).

% log_le_iff
tff(fact_1083_le__log__iff,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),aa(real,real,log2(B2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2)),X)) ) ) ) ).

% le_log_iff
tff(fact_1084_powr__non__neg,axiom,
    ! [A3: real,X: real] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),powr(real),A3),X)),zero_zero(real))) ).

% powr_non_neg
tff(fact_1085_powr__less__mono2__neg,axiom,
    ! [A3: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),powr(real),Y2),A3)),aa(real,real,aa(real,fun(real,real),powr(real),X),A3))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_1086_powr__ge__pzero,axiom,
    ! [X: real,Y2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),powr(real),X),Y2))) ).

% powr_ge_pzero
tff(fact_1087_powr__mono2,axiom,
    ! [A3: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),Y2),A3))) ) ) ) ).

% powr_mono2
tff(fact_1088_powr__less__mono,axiom,
    ! [A3: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),X),B2))) ) ) ).

% powr_less_mono
tff(fact_1089_powr__less__cancel,axiom,
    ! [X: real,A3: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),X),B2)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2)) ) ) ).

% powr_less_cancel
tff(fact_1090_powr__mono,axiom,
    ! [A3: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),X),B2))) ) ) ).

% powr_mono
tff(fact_1091_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2)) ) ) ) ).

% add_decreasing
tff(fact_1092_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% add_increasing
tff(fact_1093_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2)) ) ) ) ).

% add_decreasing2
tff(fact_1094_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% add_increasing2
tff(fact_1095_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1096_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1097_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y2 = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1098_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),zero_zero(A)))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y2 = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1099_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J2: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1100_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J2: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1101_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1102_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1103_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),zero_zero(A))) ) ) ) ).

% add_less_zeroD
tff(fact_1104_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% pos_add_strict
tff(fact_1105_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1106_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_pos_pos
tff(fact_1107_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_neg_neg
tff(fact_1108_log__def,axiom,
    ! [A3: real,X: real] : aa(real,real,log2(A3),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),A3)) ).

% log_def
tff(fact_1109_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A,J2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K)),J2)) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1110_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K))) ) ) ).

% add_le_imp_le_diff
tff(fact_1111_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_le_eq
tff(fact_1112_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2)) ) ) ).

% le_diff_eq
tff(fact_1113_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1114_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1115_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1116_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1117_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1118_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1119_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1120_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1121_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1122_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_1123_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1124_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)))) ) ).

% less_add_one
tff(fact_1125_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),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_1126_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_less_eq
tff(fact_1127_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2)) ) ) ).

% less_diff_eq
tff(fact_1128_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_1129_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_1130_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_1131_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_1132_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_1133_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B5: A,K: A,B2: A,A3: A] :
          ( ( B5 = 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),B5) = 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_1134_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_1135_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_1136_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_1137_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_1138_add__is__1,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M2 = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_1139_one__is__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) )
    <=> ( ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M2 = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_1140_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ~ ! [Q3: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),Q3)) ) ).

% less_natE
tff(fact_1141_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M2)))) ).

% less_add_Suc1
tff(fact_1142_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),I)))) ).

% less_add_Suc2
tff(fact_1143_less__iff__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
    <=> ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K3)) ) ).

% less_iff_Suc_add
tff(fact_1144_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ? [K2: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2)) ) ).

% less_imp_Suc_add
tff(fact_1145_less__imp__add__positive,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ? [K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2) = J2 ) ) ) ).

% less_imp_add_positive
tff(fact_1146_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M2: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F2,M3)),aa(nat,nat,F2,N2))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,M2)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K)))) ) ).

% mono_nat_linear_lb
tff(fact_1147_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F2: fun(A,nat),N: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [X2: A] :
            ( pp(aa(A,bool,P,X2))
           => ? [Y: A] :
                ( pp(aa(A,bool,P,Y))
                & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,Y)),aa(A,nat,F2,X2))) ) )
       => ? [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
            & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),N))) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1148_diff__add__0,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1149_less__diff__conv,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J2)) ) ).

% less_diff_conv
tff(fact_1150_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = M2 ) ) ).

% add_diff_inverse_nat
tff(fact_1151_Suc__eq__plus1,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_1152_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1153_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).

% Suc_eq_plus1_left
tff(fact_1154_odd__nonzero,axiom,
    ! [Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z) != zero_zero(int) ).

% odd_nonzero
tff(fact_1155_le__diff__conv,axiom,
    ! [J2: nat,K: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K)),I))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ).

% le_diff_conv
tff(fact_1156_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J2)) ) ) ).

% Nat.le_diff_conv2
tff(fact_1157_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J2)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1158_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1159_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I) = K )
      <=> ( J2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1160_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I3))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_ge_induct
tff(fact_1161_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I3))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_gr_induct
tff(fact_1162_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1163_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z))
    <=> ? [N5: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N5)) ) ).

% zle_iff_zadd
tff(fact_1164_minus__log__eq__powr,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),Y2),aa(real,real,log2(B2),X)) = aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_1165_minus__real__def,axiom,
    ! [X: real,Y2: real] : aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),Y2)) ).

% minus_real_def
tff(fact_1166_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_inc(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_1167_powr__less__mono2,axiom,
    ! [A3: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),Y2),A3))) ) ) ) ).

% powr_less_mono2
tff(fact_1168_powr__mono2_H,axiom,
    ! [A3: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),Y2),A3)),aa(real,real,aa(real,fun(real,real),powr(real),X),A3))) ) ) ) ).

% powr_mono2'
tff(fact_1169_powr__inj,axiom,
    ! [A3: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( ( aa(real,real,aa(real,fun(real,real),powr(real),A3),X) = aa(real,real,aa(real,fun(real,real),powr(real),A3),Y2) )
        <=> ( X = Y2 ) ) ) ) ).

% powr_inj
tff(fact_1170_gr__one__powr,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,aa(real,fun(real,real),powr(real),X),Y2))) ) ) ).

% gr_one_powr
tff(fact_1171_ge__one__powr__ge__zero,axiom,
    ! [X: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,aa(real,fun(real,real),powr(real),X),A3))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_1172_powr__mono__both,axiom,
    ! [A3: real,B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),Y2),B2))) ) ) ) ) ).

% powr_mono_both
tff(fact_1173_powr__le1,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),one_one(real))) ) ) ) ).

% powr_le1
tff(fact_1174_powr__divide,axiom,
    ! [X: real,Y2: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
       => ( aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y2)),A3) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),Y2),A3)) ) ) ) ).

% powr_divide
tff(fact_1175_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( ! [E: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y2),E))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) ) ) ).

% field_le_epsilon
tff(fact_1176_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% add_strict_increasing2
tff(fact_1177_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% add_strict_increasing
tff(fact_1178_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_pos_nonneg
tff(fact_1179_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_neg
tff(fact_1180_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_nonneg_pos
tff(fact_1181_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_neg_nonpos
tff(fact_1182_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B2)) ) ) ).

% discrete
tff(fact_1183_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ).

% zero_less_two
tff(fact_1184_abs__real__def,axiom,
    ! [A3: real] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A3) = aa(real,real,uminus_uminus(real),A3) ) )
      & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A3) = A3 ) ) ) ).

% abs_real_def
tff(fact_1185_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_1186_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_1187_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1188_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_1189_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A,C2: A,D2: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_1190_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_1191_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).

% abs_diff_le_iff
tff(fact_1192_log__ln,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),X) = aa(real,real,log2(aa(real,real,exp(real),one_one(real))),X) ).

% log_ln
tff(fact_1193_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,abs_abs(real),X)) = aa(real,real,abs_abs(real),aa(real,real,root(N),X)) ) ) ).

% real_root_abs
tff(fact_1194_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).

% abs_diff_less_iff
tff(fact_1195_powr__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [W: A,Z1: A,Z22: A] : aa(A,A,aa(A,fun(A,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),aa(A,A,aa(A,fun(A,A),powr(A),W),Z1)),aa(A,A,aa(A,fun(A,A),powr(A),W),Z22)) ) ).

% powr_diff
tff(fact_1196_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A3: A] :
        ? [B4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B4))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),A3)) ) ) ).

% ex_gt_or_lt
tff(fact_1197_nat__diff__split__asm,axiom,
    ! [P: fun(nat,bool),A3: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
            & ~ pp(aa(nat,bool,P,zero_zero(nat))) )
          | ? [D4: nat] :
              ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
              & ~ pp(aa(nat,bool,P,D4)) ) ) ) ).

% nat_diff_split_asm
tff(fact_1198_nat__diff__split,axiom,
    ! [P: fun(nat,bool),A3: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ! [D4: nat] :
            ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
           => pp(aa(nat,bool,P,D4)) ) ) ) ).

% nat_diff_split
tff(fact_1199_real__add__less__0__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y2)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(real,real,uminus_uminus(real),X))) ) ).

% real_add_less_0_iff
tff(fact_1200_real__0__less__add__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),X)),Y2)) ) ).

% real_0_less_add_iff
tff(fact_1201_real__0__le__add__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),X)),Y2)) ) ).

% real_0_le_add_iff
tff(fact_1202_real__add__le__0__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y2)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),aa(real,real,uminus_uminus(real),X))) ) ).

% real_add_le_0_iff
tff(fact_1203_less__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K)),I))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ) ).

% less_diff_conv2
tff(fact_1204_int__ops_I4_J,axiom,
    ! [A3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),one_one(int)) ).

% int_ops(4)
tff(fact_1205_int__Suc,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ).

% int_Suc
tff(fact_1206_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
    <=> ? [N5: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N5))) ) ).

% zless_iff_Suc_zadd
tff(fact_1207_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ).

% odd_less_0_iff
tff(fact_1208_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ).

% add1_zle_eq
tff(fact_1209_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)) ) ).

% zless_imp_add1_zle
tff(fact_1210_int__induct,axiom,
    ! [P: fun(int,bool),K: int,I: int] :
      ( pp(aa(int,bool,P,K))
     => ( ! [I3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I3))
           => ( pp(aa(int,bool,P,I3))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),K))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_induct
tff(fact_1211_exp__ge__add__one__self,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,exp(real),X))) ).

% exp_ge_add_one_self
tff(fact_1212_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_1213_log__base__change,axiom,
    ! [A3: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log2(B2),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log2(A3),X)),aa(real,real,log2(A3),B2)) ) ) ) ).

% log_base_change
tff(fact_1214_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),powr(real),X),aa(nat,real,semiring_1_of_nat(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N) ) ) ).

% powr_realpow
tff(fact_1215_less__log__of__power,axiom,
    ! [B2: real,N: nat,M2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)),M2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log2(B2),M2))) ) ) ).

% less_log_of_power
tff(fact_1216_log__of__power__eq,axiom,
    ! [M2: nat,B2: real,N: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M2) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log2(B2),aa(nat,real,semiring_1_of_nat(real),M2)) ) ) ) ).

% log_of_power_eq
tff(fact_1217_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X)))) ) ).

% abs_add_one_gt_zero
tff(fact_1218_root__abs__power,axiom,
    ! [N: nat,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,abs_abs(real),aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),N))) = aa(real,real,abs_abs(real),Y2) ) ) ).

% root_abs_power
tff(fact_1219_add__eq__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = N ) )
      & ( ( M2 != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = 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),M2),one_one(nat))),N)) ) ) ) ).

% add_eq_if
tff(fact_1220_nat__less__real__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),N)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M2))) ) ).

% nat_less_real_le
tff(fact_1221_nat__le__real__less,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),M2)),one_one(real)))) ) ).

% nat_le_real_less
tff(fact_1222_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z))) ) ).

% le_imp_0_less
tff(fact_1223_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).

% ln_add_one_self_le_self
tff(fact_1224_Suc__as__int,axiom,
    ! [X3: nat] : aa(nat,nat,suc,X3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),one_one(int))) ).

% Suc_as_int
tff(fact_1225_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,exp(real),X))) ) ).

% exp_ge_add_one_self_aux
tff(fact_1226_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1227_real__of__nat__div__aux,axiom,
    ! [X: nat,D2: nat] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),X)),aa(nat,real,semiring_1_of_nat(real),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),X),D2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),modulo_modulo(nat,X,D2))),aa(nat,real,semiring_1_of_nat(real),D2))) ).

% real_of_nat_div_aux
tff(fact_1228_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A3: A] : aa(A,A,aa(A,fun(A,A),powr(A),X),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),powr(A),X),A3)) ) ).

% powr_minus_divide
tff(fact_1229_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),powr(real),X),aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X) ) ) ).

% powr_neg_one
tff(fact_1230_log__divide,axiom,
    ! [A3: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
           => ( aa(real,real,log2(A3),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y2)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log2(A3),X)),aa(real,real,log2(A3),Y2)) ) ) ) ) ) ).

% log_divide
tff(fact_1231_le__log__of__power,axiom,
    ! [B2: real,N: nat,M2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)),M2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log2(B2),M2))) ) ) ).

% le_log_of_power
tff(fact_1232_log__base__pow,axiom,
    ! [A3: real,N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( aa(real,real,log2(aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),N)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log2(A3),X)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ).

% log_base_pow
tff(fact_1233_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_1234_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_1235_log__of__power__less,axiom,
    ! [M2: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),M2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log2(B2),aa(nat,real,semiring_1_of_nat(real),M2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_less
tff(fact_1236_ln__powr__bound,axiom,
    ! [X: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),A3))) ) ) ).

% ln_powr_bound
tff(fact_1237_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1238_lemma__interval,axiom,
    ! [A3: real,X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ? [D5: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
            & ! [Y: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y))),D5))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),Y))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),B2)) ) ) ) ) ) ).

% lemma_interval
tff(fact_1239_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( ( archimedean_ceiling(real,aa(real,real,log2(B2),X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),powr(real),B2),aa(nat,real,semiring_1_of_nat(real),K))),X))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),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_1240_sin__bound__lemma,axiom,
    ! [X: real,Y2: real,U: real,V2: real] :
      ( ( X = Y2 )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),U)),V2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),U)),Y2))),V2)) ) ) ).

% sin_bound_lemma
tff(fact_1241_lemma__interval__lt,axiom,
    ! [A3: real,X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ? [D5: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
            & ! [Y: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y))),D5))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Y))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),B2)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_1242_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_1243_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_1244_ceiling__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: nat] : archimedean_ceiling(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% ceiling_of_nat
tff(fact_1245_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_le_zero
tff(fact_1246_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% zero_less_ceiling
tff(fact_1247_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_less_one
tff(fact_1248_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% one_le_ceiling
tff(fact_1249_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ).

% ceiling_le_one
tff(fact_1250_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ).

% one_less_ceiling
tff(fact_1251_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_1252_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_1253_nat__ceiling__le__eq,axiom,
    ! [X: real,A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,X))),A3))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),A3))) ) ).

% nat_ceiling_le_eq
tff(fact_1254_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).

% ceiling_less_zero
tff(fact_1255_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X)) ) ) ).

% zero_le_ceiling
tff(fact_1256_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,Y2)),archimedean_ceiling(A,X))) ) ) ).

% ceiling_mono
tff(fact_1257_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2)) ) ) ).

% ceiling_less_cancel
tff(fact_1258_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_1259_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y2)))) ) ).

% ceiling_add_le
tff(fact_1260_real__nat__ceiling__ge,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),aa(int,nat,nat2,archimedean_ceiling(real,X))))) ).

% real_nat_ceiling_ge
tff(fact_1261_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs2)))
             => pp(aa(list(A),bool,P,Ys)) )
         => pp(aa(list(A),bool,P,Xs2)) )
     => pp(aa(list(A),bool,P,Xs)) ) ).

% length_induct
tff(fact_1262_eq__diff__eq_H,axiom,
    ! [X: real,Y2: real,Z: real] :
      ( ( X = aa(real,real,aa(real,fun(real,real),minus_minus(real),Y2),Z) )
    <=> ( Y2 = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Z) ) ) ).

% eq_diff_eq'
tff(fact_1263_powr__int,axiom,
    ! [X: real,I: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
         => ( aa(real,real,aa(real,fun(real,real),powr(real),X),aa(int,real,ring_1_of_int(real),I)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(int,nat,nat2,I)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
         => ( aa(real,real,aa(real,fun(real,real),powr(real),X),aa(int,real,ring_1_of_int(real),I)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I)))) ) ) ) ) ).

% powr_int
tff(fact_1264_log__minus__eq__powr,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log2(B2),X)),Y2) = aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),aa(real,real,aa(real,fun(real,real),powr(real),B2),aa(real,real,uminus_uminus(real),Y2)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_1265_log__base__root,axiom,
    ! [N: nat,B2: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
       => ( aa(real,real,log2(aa(real,real,root(N),B2)),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log2(B2),X)) ) ) ) ).

% log_base_root
tff(fact_1266_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tanh(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% tanh_altdef
tff(fact_1267_triangle__Suc,axiom,
    ! [N: nat] : nat_triangle(aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(N)),aa(nat,nat,suc,N)) ).

% triangle_Suc
tff(fact_1268_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),N)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),N)),one_one(A))))
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_1269_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_1270_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),modulo_modulo(int,N,K)))
      <=> ! [I2: int,J3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I2),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_1271_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,tanh(real),X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% tanh_real_zero_iff
tff(fact_1272_tanh__real__less__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tanh(real),X)),aa(real,real,tanh(real),Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ).

% tanh_real_less_iff
tff(fact_1273_tanh__real__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tanh(real),X)),aa(real,real,tanh(real),Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ).

% tanh_real_le_iff
tff(fact_1274_of__int__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W: int,Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),W) = aa(int,A,ring_1_of_int(A),Z) )
        <=> ( W = Z ) ) ) ).

% of_int_eq_iff
tff(fact_1275_tanh__real__abs,axiom,
    ! [X: real] : aa(real,real,tanh(real),aa(real,real,abs_abs(real),X)) = aa(real,real,abs_abs(real),aa(real,real,tanh(real),X)) ).

% tanh_real_abs
tff(fact_1276_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_1277_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_1278_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_1279_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_1280_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_1281_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_1282_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_1283_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_1284_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_1285_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_1286_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_1287_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_1288_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_1289_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_1290_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_1291_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_1292_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tanh(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% tanh_real_neg_iff
tff(fact_1293_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,tanh(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% tanh_real_pos_iff
tff(fact_1294_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tanh(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% tanh_real_nonpos_iff
tff(fact_1295_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,tanh(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% tanh_real_nonneg_iff
tff(fact_1296_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_1297_tanh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tanh_0
tff(fact_1298_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)) = X )
        <=> ? [N5: int] : X = aa(int,A,ring_1_of_int(A),N5) ) ) ).

% of_int_ceiling_cancel
tff(fact_1299_ceiling__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archimedean_ceiling(A,aa(int,A,ring_1_of_int(A),Z)) = Z ) ).

% ceiling_of_int
tff(fact_1300_tanh__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tanh(A),aa(A,A,uminus_uminus(A),X)) = aa(A,A,uminus_uminus(A),aa(A,A,tanh(A),X)) ) ).

% tanh_minus
tff(fact_1301_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_1302_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_1303_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_1304_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_1305_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_1306_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Y2)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y2 = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1307_mult__divide__mult__cancel__left__if,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)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),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) ) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1308_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_1309_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_1310_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_1311_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_1312_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_1313_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_1314_div__mult__mult1__if,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)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),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_if
tff(fact_1315_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_1316_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_1317_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_1318_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_1319_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_1320_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_1321_of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = zero_zero(A) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_eq_0_iff
tff(fact_1322_of__int__0__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( zero_zero(A) = aa(int,A,ring_1_of_int(A),Z) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_0_eq_iff
tff(fact_1323_of__int__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),zero_zero(int)) = zero_zero(A) ) ) ).

% of_int_0
tff(fact_1324_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ) ).

% of_int_le_iff
tff(fact_1325_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% of_int_less_iff
tff(fact_1326_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_1327_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_1328_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_1329_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_1330_of__int__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),Z)) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_minus
tff(fact_1331_of__int__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),minus_minus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_diff
tff(fact_1332_of__int__of__nat__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% of_int_of_nat_eq
tff(fact_1333_of__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int] : aa(int,A,ring_1_of_int(A),aa(int,int,abs_abs(int),X)) = aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),X)) ) ).

% of_int_abs
tff(fact_1334_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,N: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),Z)),N) ) ).

% of_int_power
tff(fact_1335_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W: nat,X: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) = aa(int,A,ring_1_of_int(A),X) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) = X ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_1336_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: int,B2: int,W: nat] :
          ( ( aa(int,A,ring_1_of_int(A),X) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) )
        <=> ( X = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_1337_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_1338_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_1339_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_1340_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_1341_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_1342_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_1343_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: 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))),N)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),A3)) = A3 ) ).

% left_minus_one_mult_self
tff(fact_1344_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_1345_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),Z) ) ).

% ceiling_add_of_int
tff(fact_1346_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),Z) ) ).

% ceiling_diff_of_int
tff(fact_1347_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int))) ) ) ).

% of_int_le_0_iff
tff(fact_1348_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) ) ) ).

% of_int_0_le_iff
tff(fact_1349_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ) ).

% of_int_less_0_iff
tff(fact_1350_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ) ).

% of_int_0_less_iff
tff(fact_1351_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),one_one(int))) ) ) ).

% of_int_le_1_iff
tff(fact_1352_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z)) ) ) ).

% of_int_1_le_iff
tff(fact_1353_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),one_one(int))) ) ) ).

% of_int_less_1_iff
tff(fact_1354_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z)) ) ) ).

% of_int_1_less_iff
tff(fact_1355_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),X)) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_1356_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W))) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_1357_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W))) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_1358_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),X)) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_1359_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ) ).

% of_nat_nat
tff(fact_1360_neq__if__length__neq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) != aa(list(A),nat,size_size(list(A)),Ys2) )
     => ( Xs != Ys2 ) ) ).

% neq_if_length_neq
tff(fact_1361_Ex__list__of__length,axiom,
    ! [A: $tType,N: nat] :
    ? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = N ).

% Ex_list_of_length
tff(fact_1362_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_1363_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_1364_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_1365_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_1366_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int,Y2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),X)),Y2) = aa(A,A,aa(A,fun(A,A),times_times(A),Y2),aa(int,A,ring_1_of_int(A),X)) ) ).

% mult_of_int_commute
tff(fact_1367_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).

% ex_le_of_int
tff(fact_1368_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),X)) ) ).

% ex_of_int_less
tff(fact_1369_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).

% ex_less_of_int
tff(fact_1370_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_1371_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_1372_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_1373_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_1374_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_1375_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_1376_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_1377_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_1378_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_1379_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_1380_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_1381_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_1382_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E2: 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),E2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),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)),E2)),C2) ) ).

% combine_common_factor
tff(fact_1383_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,bool),D6: A,Q: fun(A,bool)] :
          ( ! [X2: A,K2: A] :
              ( pp(aa(A,bool,P,X2))
            <=> pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D6)))) )
         => ( ! [X2: A,K2: A] :
                ( pp(aa(A,bool,Q,X2))
              <=> pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D6)))) )
           => ! [X3: A,K4: A] :
                ( ( pp(aa(A,bool,P,X3))
                  & pp(aa(A,bool,Q,X3)) )
              <=> ( pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D6))))
                  & pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D6)))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1384_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,bool),D6: A,Q: fun(A,bool)] :
          ( ! [X2: A,K2: A] :
              ( pp(aa(A,bool,P,X2))
            <=> pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D6)))) )
         => ( ! [X2: A,K2: A] :
                ( pp(aa(A,bool,Q,X2))
              <=> pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D6)))) )
           => ! [X3: A,K4: A] :
                ( ( pp(aa(A,bool,P,X3))
                  | pp(aa(A,bool,Q,X3)) )
              <=> ( pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D6))))
                  | pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D6)))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1385_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_1386_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_1387_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_1388_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_1389_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_1390_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y2: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),W)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z)) ) ).

% divide_divide_times_eq
tff(fact_1391_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y2: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),W)) ) ).

% times_divide_times_eq
tff(fact_1392_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_1393_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_1394_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y2) = aa(A,A,aa(A,fun(A,A),times_times(A),Y2),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_1395_times__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_1396_times__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),times_times(int),zero_zero(int)),L) = zero_zero(int) ).

% times_int_code(2)
tff(fact_1397_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_1398_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(2)
tff(fact_1399_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(1)
tff(fact_1400_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(3)
tff(fact_1401_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(4)
tff(fact_1402_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),aa(A,A,exp(A),A4)),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,exp(A),A4)) ) ).

% exp_times_arg_commute
tff(fact_1403_powr__powr,axiom,
    ! [X: real,A3: real,B2: real] : aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),B2) = aa(real,real,aa(real,fun(real,real),powr(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),A3),B2)) ).

% powr_powr
tff(fact_1404_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_1405_artanh__tanh__real,axiom,
    ! [X: real] : aa(real,real,artanh(real),aa(real,real,tanh(real),X)) = X ).

% artanh_tanh_real
tff(fact_1406_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),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_1407_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ).

% le_of_int_ceiling
tff(fact_1408_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1409_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1410_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1411_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1412_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1413_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1414_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1415_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1416_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1417_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1418_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1419_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1420_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1421_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1422_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_1423_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),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_1424_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),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_1425_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1426_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1427_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1428_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1429_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1430_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1431_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1432_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1433_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1434_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1435_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos
tff(fact_1436_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1437_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).

% mult_pos_neg2
tff(fact_1438_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_pos_pos
tff(fact_1439_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg
tff(fact_1440_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_neg_pos
tff(fact_1441_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1442_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),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_1443_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_neg_neg
tff(fact_1444_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M2: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M2),N))) ) ) ) ).

% less_1_mult
tff(fact_1445_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y2: A,Z: A,X: A,W: A] :
          ( ( Y2 != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y2) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1446_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 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq
tff(fact_1447_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) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq
tff(fact_1448_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_1449_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_1450_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_1451_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_1452_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E2: 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),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),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)),E2)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_1453_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E2: 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),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),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)),E2)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_1454_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Y2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) ) ).

% square_diff_square_factored
tff(fact_1455_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [X: A,Y2: 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),X),Y2)),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),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y2),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3)),B2)) ) ).

% mult_diff_mult
tff(fact_1456_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),X) = one_one(A) )
        <=> ( ( X = one_one(A) )
            | ( X = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_1457_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y2) = one_one(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),N)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_1458_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_Suc
tff(fact_1459_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),A3) ) ).

% power_Suc2
tff(fact_1460_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),B2)),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1461_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M2: nat,N: 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),M2)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M2))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% div_mult2_eq'
tff(fact_1462_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y2)) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) ) ).

% mult_exp_exp
tff(fact_1463_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y2) = aa(A,A,aa(A,fun(A,A),times_times(A),Y2),X) )
         => ( aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y2)) ) ) ) ).

% exp_add_commuting
tff(fact_1464_zmult__zless__mono2,axiom,
    ! [I: int,J2: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),J2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J2))) ) ) ).

% zmult_zless_mono2
tff(fact_1465_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),powr(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),powr(A),X),A3)),aa(A,A,aa(A,fun(A,A),powr(A),X),B2)) ) ).

% powr_add
tff(fact_1466_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),U),U))),aa(real,real,aa(real,fun(real,real),times_times(real),X),X))) ).

% real_minus_mult_self_le
tff(fact_1467_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M2: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M2),N) = one_one(int) )
     => ( ( M2 = one_one(int) )
        | ( M2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_1468_zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M2),N) = one_one(int) )
    <=> ( ( ( M2 = one_one(int) )
          & ( N = one_one(int) ) )
        | ( ( M2 = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_1469_abs__zmult__eq__1,axiom,
    ! [M2: int,N: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M2),N)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),M2) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_1470_zmod__eq__0D,axiom,
    ! [M2: int,D2: int] :
      ( ( modulo_modulo(int,M2,D2) = zero_zero(int) )
     => ? [Q3: int] : M2 = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q3) ) ).

% zmod_eq_0D
tff(fact_1471_zmod__eq__0__iff,axiom,
    ! [M2: int,D2: int] :
      ( ( modulo_modulo(int,M2,D2) = zero_zero(int) )
    <=> ? [Q4: int] : M2 = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q4) ) ).

% zmod_eq_0_iff
tff(fact_1472_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q5)))),Q5))) ) ) ).

% ceiling_divide_upper
tff(fact_1473_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tanh(real),X)),one_one(real))) ).

% tanh_real_lt_1
tff(fact_1474_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q5))
         => pp(aa(A,bool,aa(A,fun(A,bool),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),Q5)))),one_one(A))),Q5)),P2)) ) ) ).

% ceiling_divide_lower
tff(fact_1475_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),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_1476_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),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_1477_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1478_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1479_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1480_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1481_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1482_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),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_1483_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1484_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1485_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),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_1486_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1487_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1488_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1489_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% ceiling_le_iff
tff(fact_1490_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),A3)) ) ) ).

% ceiling_le
tff(fact_1491_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),X)) ) ) ).

% less_ceiling_iff
tff(fact_1492_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),A3)) ) ) ) ).

% mult_left_le
tff(fact_1493_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),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_1494_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y2)),X)) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1495_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),X)),X)) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1496_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Y2)))) ) ).

% sum_squares_ge_zero
tff(fact_1497_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Y2))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y2 = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1498_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Y2))),zero_zero(A))) ) ).

% not_sum_squares_lt_zero
tff(fact_1499_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Y2))))
        <=> ( ( X != zero_zero(A) )
            | ( Y2 != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1500_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1501_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),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_1502_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),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_1503_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y2: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y2)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2))) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1504_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y2: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),Z)) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1505_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ).

% pos_less_divide_eq
tff(fact_1506_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% pos_divide_less_eq
tff(fact_1507_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% neg_less_divide_eq
tff(fact_1508_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ).

% neg_divide_less_eq
tff(fact_1509_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq
tff(fact_1510_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% divide_less_eq
tff(fact_1511_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E2)),C2)),D2)) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1512_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E2)),D2))) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1513_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z)),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z)),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_1514_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = zero_zero(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)) = A3 ) )
          & ( ( Z != zero_zero(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)) = 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_1515_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y2: A,Z: A,X: A,W: A] :
          ( ( Y2 != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1516_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y2: A,X: A,Z: A] :
          ( ( Y2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y2))),Y2) ) ) ) ).

% add_frac_num
tff(fact_1517_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y2: A,Z: A,X: A] :
          ( ( Y2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y2))),Y2) ) ) ) ).

% add_num_frac
tff(fact_1518_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y2: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),Y2)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1519_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y2: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),Y2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1520_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E2)),D2))) ) ) ).

% less_add_iff2
tff(fact_1521_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E2)),C2)),D2)) ) ) ).

% less_add_iff1
tff(fact_1522_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y2: 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),X),Z)),Y2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1523_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y2: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y2),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),X),Z)),Y2)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1524_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y2: A,Z: A,X: A,W: A] :
          ( ( Y2 != 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),X),Y2)),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),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1525_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = zero_zero(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)) = A3 ) )
          & ( ( Z != zero_zero(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)) = 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_1526_real__of__int__div4,axiom,
    ! [N: int,X: int] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),X))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),X)))) ).

% real_of_int_div4
tff(fact_1527_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),X))) ) ) ).

% ex_less_of_nat_mult
tff(fact_1528_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)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,uminus_uminus(A),B2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_1529_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 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_1530_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_1531_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_1532_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))) ) ) ).

% power_less_power_Suc
tff(fact_1533_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),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),N)))) ) ) ).

% power_gt1_lemma
tff(fact_1534_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1535_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y2)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),X)) ) ) ) ).

% abs_mult_pos
tff(fact_1536_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_1537_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_minus
tff(fact_1538_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_1539_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_1540_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_1541_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_1542_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_1543_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_1544_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_1545_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_1546_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_1547_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_1548_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_1549_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_1550_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_1551_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,N: nat] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),X)),N) ) ).

% exp_of_nat2_mult
tff(fact_1552_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,X: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),X)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),X)),N) ) ).

% exp_of_nat_mult
tff(fact_1553_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ! [Y: real] :
        ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),X))) ) ).

% reals_Archimedean3
tff(fact_1554_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M2))
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M2),N) = one_one(int) )
      <=> ( ( M2 = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1555_powr__mult,axiom,
    ! [X: real,Y2: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
       => ( aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y2)),A3) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),powr(real),X),A3)),aa(real,real,aa(real,fun(real,real),powr(real),Y2),A3)) ) ) ) ).

% powr_mult
tff(fact_1556_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X2: int,K2: int] :
            ( pp(aa(int,bool,P1,X2))
          <=> pp(aa(int,bool,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),Z4))
             => ( pp(aa(int,bool,P,X2))
              <=> pp(aa(int,bool,P1,X2)) ) )
         => ( ? [X_12: int] : pp(aa(int,bool,P1,X_12))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_1557_plusinfinity,axiom,
    ! [D2: int,P3: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X2: int,K2: int] :
            ( pp(aa(int,bool,P3,X2))
          <=> pp(aa(int,bool,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),X2))
             => ( pp(aa(int,bool,P,X2))
              <=> pp(aa(int,bool,P3,X2)) ) )
         => ( ? [X_12: int] : pp(aa(int,bool,P3,X_12))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_1558_zdiv__zmult2__eq,axiom,
    ! [C2: int,A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_1559_divide__powr__uminus,axiom,
    ! [A3: real,B2: real,C2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),A3),aa(real,real,aa(real,fun(real,real),powr(real),B2),C2)) = aa(real,real,aa(real,fun(real,real),times_times(real),A3),aa(real,real,aa(real,fun(real,real),powr(real),B2),aa(real,real,uminus_uminus(real),C2))) ).

% divide_powr_uminus
tff(fact_1560_ln__powr,axiom,
    ! [X: real,Y2: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),powr(real),X),Y2)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y2),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_powr
tff(fact_1561_log__powr,axiom,
    ! [X: real,B2: real,Y2: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),powr(real),X),Y2)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y2),aa(real,real,log2(B2),X)) ) ) ).

% log_powr
tff(fact_1562_div__mod__decomp__int,axiom,
    ! [A4: int,N: 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),N)),N)),modulo_modulo(int,A4,N)) ).

% div_mod_decomp_int
tff(fact_1563_tanh__real__gt__neg1,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),X))) ).

% tanh_real_gt_neg1
tff(fact_1564_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% of_int_nonneg
tff(fact_1565_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% of_int_leD
tff(fact_1566_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% of_int_pos
tff(fact_1567_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% of_int_lessD
tff(fact_1568_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X2)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int)))))
          & ! [Y: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),X))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y),one_one(int))))) )
             => ( Y = X2 ) ) ) ) ).

% floor_exists1
tff(fact_1569_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int))))) ) ) ).

% floor_exists
tff(fact_1570_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_1571_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_1572_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),X)) ) ) ).

% of_nat_less_of_int_iff
tff(fact_1573_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( ! [Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),one_one(A)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),X)),Y2)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) ) ) ).

% field_le_mult_one_interval
tff(fact_1574_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1575_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1576_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1577_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1578_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1579_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1580_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1581_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1582_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% divide_le_eq
tff(fact_1583_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq
tff(fact_1584_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),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_1585_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ).

% neg_divide_le_eq
tff(fact_1586_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% neg_le_divide_eq
tff(fact_1587_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% pos_divide_le_eq
tff(fact_1588_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ).

% pos_le_divide_eq
tff(fact_1589_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y2: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),Z)) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1590_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y2: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y2)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2))) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1591_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),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_1592_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A3: A,Y2: A,U: A,V2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y2))),A3)) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1593_int__le__real__less,axiom,
    ! [N: int,M2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),M2))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(int,real,ring_1_of_int(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),M2)),one_one(real)))) ) ).

% int_le_real_less
tff(fact_1594_int__less__real__le,axiom,
    ! [N: int,M2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),M2))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))),aa(int,real,ring_1_of_int(real),M2))) ) ).

% int_less_real_le
tff(fact_1595_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y2: A,Z: A,X: A,W: A] :
          ( ( Y2 != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),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),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z))),zero_zero(A))) ) ) ) ) ).

% frac_le_eq
tff(fact_1596_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y2: A,Z: A,X: A,W: A] :
          ( ( Y2 != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),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),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z))),zero_zero(A))) ) ) ) ) ).

% frac_less_eq
tff(fact_1597_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).

% power_Suc_less
tff(fact_1598_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1599_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1600_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1601_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1602_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_1603_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_1604_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z))),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z))),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1605_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y2: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z))),Y2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1606_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: 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),A3),Z))),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( 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),A3),Z))),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_1607_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: 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),A3),Z)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( 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),A3),Z)),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_1608_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y2: 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),X),Z))),Y2) = 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),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1609_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_1610_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M2: nat,N: nat] : modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A3,aa(nat,A,semiring_1_of_nat(A),M2))) ) ).

% mod_mult2_eq'
tff(fact_1611_real__of__int__div__aux,axiom,
    ! [X: int,D2: int] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),X)),aa(int,real,ring_1_of_int(real),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),X),D2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),modulo_modulo(int,X,D2))),aa(int,real,ring_1_of_int(real),D2))) ).

% real_of_int_div_aux
tff(fact_1612_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J2: int,K: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J2))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1613_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X2: int] :
            ( pp(aa(int,bool,P,X2))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X3: int] :
              ( pp(aa(int,bool,P,X3))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1614_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q6: int,R4: int,Q5: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R4))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q5),Q6)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1615_unique__quotient__lemma,axiom,
    ! [B2: int,Q6: int,R4: int,Q5: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R4))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q6),Q5)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1616_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q5: int,R: int,B6: int,Q6: 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),Q5)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4)),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R4))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B6),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q6),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1617_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q5: int,R: int,B6: int,Q6: 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),Q5)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B6))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B6),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q5),Q6)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1618_q__pos__lemma,axiom,
    ! [B6: int,Q6: int,R4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B6))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q6)) ) ) ) ).

% q_pos_lemma
tff(fact_1619_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X2: int] :
            ( pp(aa(int,bool,P,X2))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D2))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X3: int] :
              ( pp(aa(int,bool,P,X3))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1620_ln__mult,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y2)) ) ) ) ).

% ln_mult
tff(fact_1621_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A,A3: A] :
          ( ( ( X = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),powr(A),X),A3) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),powr(A),X),A3) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,ln_ln(A),X))) ) ) ) ) ).

% powr_def
tff(fact_1622_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P,archimedean_ceiling(A,T2)))
        <=> ! [I2: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A))),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),aa(int,A,ring_1_of_int(A),I2))) )
             => pp(aa(int,bool,P,I2)) ) ) ) ).

% ceiling_split
tff(fact_1623_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archimedean_ceiling(A,X) = A3 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),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))),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3))) ) ) ) ).

% ceiling_eq_iff
tff(fact_1624_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z)))
           => ( archimedean_ceiling(A,X) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_1625_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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,X))),one_one(A))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ) ).

% ceiling_correct
tff(fact_1626_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),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_1627_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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))),X)) ) ) ).

% le_ceiling_iff
tff(fact_1628_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A3: A,Y2: A,U: A,V2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y2))),A3)) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1629_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_1630_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_1631_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1632_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1633_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1634_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1635_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R: A,S: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R),S))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(A,A,aa(A,fun(A,A),minus_minus(A),V2),U))),S))),V2)) ) ) ) ) ).

% scaling_mono
tff(fact_1636_real__of__int__div2,axiom,
    ! [N: int,X: int] : pp(aa(real,bool,aa(real,fun(real,bool),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),N)),aa(int,real,ring_1_of_int(real),X))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),X))))) ).

% real_of_int_div2
tff(fact_1637_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1638_real__of__int__div3,axiom,
    ! [N: int,X: int] : pp(aa(real,bool,aa(real,fun(real,bool),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),N)),aa(int,real,ring_1_of_int(real),X))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),X)))),one_one(real))) ).

% real_of_int_div3
tff(fact_1639_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [M2: nat,P2: A] :
          ( ( ( M2 = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M2) = one_one(A) ) )
          & ( ( M2 != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M2) = 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),M2),one_one(nat)))) ) ) ) ) ).

% power_eq_if
tff(fact_1640_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).

% power_minus_mult
tff(fact_1641_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C2))
       => ( ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M3))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M3)),X)),C2)) )
         => ( X = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1642_powr__mult__base,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),X),aa(real,real,aa(real,fun(real,real),powr(real),X),Y2)) = aa(real,real,aa(real,fun(real,real),powr(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y2)) ) ) ).

% powr_mult_base
tff(fact_1643_log__mult,axiom,
    ! [A3: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
           => ( aa(real,real,log2(A3),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log2(A3),X)),aa(real,real,log2(A3),Y2)) ) ) ) ) ) ).

% log_mult
tff(fact_1644_split__zdiv,axiom,
    ! [P: fun(int,bool),N: int,K: int] :
      ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,zero_zero(int))) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I2: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
             => pp(aa(int,bool,P,I2)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I2: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
             => pp(aa(int,bool,P,I2)) ) ) ) ) ).

% split_zdiv
tff(fact_1645_int__div__neg__eq,axiom,
    ! [A3: int,B2: int,Q5: 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),Q5)),R) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q5 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1646_int__div__pos__eq,axiom,
    ! [A3: int,B2: int,Q5: 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),Q5)),R) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q5 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1647_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_realpow
tff(fact_1648_log__nat__power,axiom,
    ! [X: real,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log2(B2),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log2(B2),X)) ) ) ).

% log_nat_power
tff(fact_1649_int__mod__pos__eq,axiom,
    ! [A3: int,B2: int,Q5: 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),Q5)),R) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
         => ( modulo_modulo(int,A3,B2) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1650_int__mod__neg__eq,axiom,
    ! [A3: int,B2: int,Q5: 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),Q5)),R) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R))
         => ( modulo_modulo(int,A3,B2) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1651_split__zmod,axiom,
    ! [P: fun(int,bool),N: int,K: int] :
      ( pp(aa(int,bool,P,modulo_modulo(int,N,K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,N)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I2: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
             => pp(aa(int,bool,P,J3)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I2: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
             => pp(aa(int,bool,P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_1652_zmod__zmult2__eq,axiom,
    ! [C2: int,A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_1653_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K) = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K) = aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K)) ) ) ) ) ).

% of_int_of_nat
tff(fact_1654_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),X)),one_one(real))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),one_one(real))),N))) ) ).

% linear_plus_1_le_power
tff(fact_1655_ln__powr__bound2,axiom,
    ! [X: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,ln_ln(real),X)),A3)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),powr(real),A3),A3)),X))) ) ) ).

% ln_powr_bound2
tff(fact_1656_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_1657_log__eq__div__ln__mult__log,axiom,
    ! [A3: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
         => ( ( B2 != one_one(real) )
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
             => ( aa(real,real,log2(A3),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(real,real,ln_ln(real),A3))),aa(real,real,log2(B2),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_1658_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Z))),one_one(int))),D2)))) ) ).

% incr_lemma
tff(fact_1659_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Z))),one_one(int))),D2))),Z)) ) ).

% decr_lemma
tff(fact_1660_log__add__eq__powr,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log2(B2),X)),Y2) = aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_1661_add__log__eq__powr,axiom,
    ! [B2: real,X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y2),aa(real,real,log2(B2),X)) = aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),powr(real),B2),Y2)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_1662_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),modulo_modulo(int,N,K)))
      <=> ! [I2: int,J3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I2),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_1663_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1664_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1665_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2)) ) ) ) ).

% mult_less_iff1
tff(fact_1666_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
         => ( aa(real,real,aa(real,fun(real,real),powr(real),X),aa(int,real,ring_1_of_int(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(int,nat,nat2,N)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
         => ( aa(real,real,aa(real,fun(real,real),powr(real),X),aa(int,real,ring_1_of_int(real),N)) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_1667_floor__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log2(B2),X)) = K )
        <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),powr(real),B2),aa(int,real,ring_1_of_int(real),K))),X))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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_1668_arctan__add,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),Y2)),one_one(real)))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y2)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y2)),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),X),Y2)))) ) ) ) ).

% arctan_add
tff(fact_1669_split__root,axiom,
    ! [P: fun(real,bool),N: nat,X: real] :
      ( pp(aa(real,bool,P,aa(real,real,root(N),X)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(real,bool,P,zero_zero(real))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ! [Y4: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y4)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y4)),N)) = X )
             => pp(aa(real,bool,P,Y4)) ) ) ) ) ).

% split_root
tff(fact_1670_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,cosh(A),X) != zero_zero(A) )
         => ( ( aa(A,A,cosh(A),Y2) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y2)))) ) ) ) ) ).

% tanh_add
tff(fact_1671_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A3)) = aa(A,A,sgn_sgn(A),A3) ) ).

% sgn_sgn
tff(fact_1672_cosh__real__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa(real,real,cosh(real),X) = aa(real,real,cosh(real),Y2) )
    <=> ( aa(real,real,abs_abs(real),X) = aa(real,real,abs_abs(real),Y2) ) ) ).

% cosh_real_eq_iff
tff(fact_1673_cosh__real__abs,axiom,
    ! [X: real] : aa(real,real,cosh(real),aa(real,real,abs_abs(real),X)) = aa(real,real,cosh(real),X) ).

% cosh_real_abs
tff(fact_1674_tanh__real__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa(real,real,tanh(real),X) = aa(real,real,tanh(real),Y2) )
    <=> ( X = Y2 ) ) ).

% tanh_real_eq_iff
tff(fact_1675_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_1676_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_1677_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] :
          ( ( aa(A,A,inverse_inverse(A),X) = one_one(A) )
        <=> ( X = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_1678_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_1679_mult__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = zero_zero(nat) )
    <=> ( ( M2 = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_1680_mult__0__right,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_1681_mult__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( M2 = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_1682_mult__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K) )
    <=> ( ( M2 = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_1683_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_1684_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_1685_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_1686_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_1687_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A3: A,B2: A] : aa(A,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),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_divide
tff(fact_1688_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A3)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_1689_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = one_one(nat) )
    <=> ( ( M2 = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1690_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) )
    <=> ( ( M2 = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1691_cosh__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,cosh(A),aa(A,A,uminus_uminus(A),X)) = aa(A,A,cosh(A),X) ) ).

% cosh_minus
tff(fact_1692_floor__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archim6421214686448440834_floor(A,aa(int,A,ring_1_of_int(A),Z)) = Z ) ).

% floor_of_int
tff(fact_1693_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) = X )
        <=> ? [N5: int] : X = aa(int,A,ring_1_of_int(A),N5) ) ) ).

% of_int_floor_cancel
tff(fact_1694_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,arctan,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% arctan_eq_zero_iff
tff(fact_1695_arctan__zero__zero,axiom,
    aa(real,real,arctan,zero_zero(real)) = zero_zero(real) ).

% arctan_zero_zero
tff(fact_1696_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_1697_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_1698_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_1699_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_1700_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% inverse_negative_iff_negative
tff(fact_1701_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% inverse_positive_iff_positive
tff(fact_1702_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,sgn_sgn(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% sgn_less
tff(fact_1703_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% sgn_greater
tff(fact_1704_one__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) )
    <=> ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_1705_mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_1706_mult__less__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% mult_less_cancel2
tff(fact_1707_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_1708_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,sgn_sgn(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,sgn_sgn(A),B2)) ) ).

% divide_sgn
tff(fact_1709_mult__Suc__right,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) ).

% mult_Suc_right
tff(fact_1710_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_1711_cosh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,cosh(A),zero_zero(A)) = one_one(A) ) ) ).

% cosh_0
tff(fact_1712_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_1713_floor__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: nat] : archim6421214686448440834_floor(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% floor_of_nat
tff(fact_1714_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,arctan,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% zero_less_arctan_iff
tff(fact_1715_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% arctan_less_zero_iff
tff(fact_1716_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arctan,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% arctan_le_zero_iff
tff(fact_1717_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arctan,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% zero_le_arctan_iff
tff(fact_1718_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_1719_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_1720_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_1721_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_1722_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_1723_one__le__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),N)) ) ) ).

% one_le_mult_iff
tff(fact_1724_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_1725_mult__le__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% mult_le_cancel2
tff(fact_1726_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),N) = M2 ) ) ).

% div_mult_self_is_m
tff(fact_1727_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2)),N) = M2 ) ) ).

% div_mult_self1_is_m
tff(fact_1728_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M2: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)),M2)),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% Suc_mod_mult_self4
tff(fact_1729_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M2: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),M2)),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% Suc_mod_mult_self3
tff(fact_1730_Suc__mod__mult__self2,axiom,
    ! [M2: nat,N: nat,K: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% Suc_mod_mult_self2
tff(fact_1731_Suc__mod__mult__self1,axiom,
    ! [M2: nat,K: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% Suc_mod_mult_self1
tff(fact_1732_floor__uminus__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,uminus_uminus(int),Z) ) ).

% floor_uminus_of_int
tff(fact_1733_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),Z) ) ).

% floor_diff_of_int
tff(fact_1734_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_1735_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ) ).

% zero_le_floor
tff(fact_1736_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A))) ) ) ).

% floor_less_zero
tff(fact_1737_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% zero_less_floor
tff(fact_1738_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_le_zero
tff(fact_1739_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% one_le_floor
tff(fact_1740_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_less_one
tff(fact_1741_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ).

% floor_diff_one
tff(fact_1742_arctan__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa(real,real,arctan,X) = aa(real,real,arctan,Y2) )
    <=> ( X = Y2 ) ) ).

% arctan_eq_iff
tff(fact_1743_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_1744_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_1745_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_1746_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_1747_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_1748_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_1749_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_1750_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B2: A] : aa(A,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),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_mult
tff(fact_1751_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,sgn_sgn(A),A3) ) ) ) ).

% same_sgn_sgn_add
tff(fact_1752_cosh__real__nonzero,axiom,
    ! [X: real] : aa(real,real,cosh(real),X) != zero_zero(real) ).

% cosh_real_nonzero
tff(fact_1753_Suc__mult__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N) )
    <=> ( M2 = N ) ) ).

% Suc_mult_cancel1
tff(fact_1754_mult__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% mult_0
tff(fact_1755_arctan__monotone,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y2))) ) ).

% arctan_monotone
tff(fact_1756_arctan__less__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ).

% arctan_less_iff
tff(fact_1757_mult__le__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J2))) ) ).

% mult_le_mono2
tff(fact_1758_mult__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),K))) ) ).

% mult_le_mono1
tff(fact_1759_mult__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),L))) ) ) ).

% mult_le_mono
tff(fact_1760_le__square,axiom,
    ! [M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),M2))) ).

% le_square
tff(fact_1761_le__cube,axiom,
    ! [M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),M2)))) ).

% le_cube
tff(fact_1762_arctan__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ).

% arctan_le_iff
tff(fact_1763_arctan__monotone_H,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y2))) ) ).

% arctan_monotone'
tff(fact_1764_add__mult__distrib2,axiom,
    ! [K: nat,M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% add_mult_distrib2
tff(fact_1765_add__mult__distrib,axiom,
    ! [M2: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% add_mult_distrib
tff(fact_1766_diff__mult__distrib2,axiom,
    ! [K: nat,M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% diff_mult_distrib2
tff(fact_1767_diff__mult__distrib,axiom,
    ! [M2: nat,N: 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),M2),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% diff_mult_distrib
tff(fact_1768_nat__mult__1__right,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).

% nat_mult_1_right
tff(fact_1769_nat__mult__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).

% nat_mult_1
tff(fact_1770_div__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),Q5) ).

% div_mult2_eq
tff(fact_1771_arctan__minus,axiom,
    ! [X: real] : aa(real,real,arctan,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,arctan,X)) ).

% arctan_minus
tff(fact_1772_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y2))) ) ) ).

% floor_mono
tff(fact_1773_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)) ) ).

% of_int_floor_le
tff(fact_1774_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% inverse_less_imp_less
tff(fact_1775_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% less_imp_inverse_less
tff(fact_1776_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_1777_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_1778_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_1779_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_1780_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))) ) ) ).

% negative_imp_inverse_negative
tff(fact_1781_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ).

% positive_imp_inverse_positive
tff(fact_1782_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2)) ) ) ).

% floor_less_cancel
tff(fact_1783_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_1784_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_1785_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_1786_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_1787_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_1788_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_1789_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_1790_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)) ) ).

% power_mult_power_inverse_commute
tff(fact_1791_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)) ) ).

% power_mult_inverse_distrib
tff(fact_1792_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: nat,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_1793_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_1794_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) != aa(A,A,sgn_sgn(A),A3) )
         => ( ( aa(A,A,sgn_sgn(A),A3) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_1795_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: int,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))) ) ).

% mult_inverse_of_int_commute
tff(fact_1796_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_1797_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_1798_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)),aa(A,A,sgn_sgn(A),A3)) = A3 ) ).

% abs_mult_sgn
tff(fact_1799_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,abs_abs(A),A3)) = A3 ) ).

% sgn_mult_abs
tff(fact_1800_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),X)),aa(A,A,abs_abs(A),X)) = X ) ).

% mult_sgn_abs
tff(fact_1801_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,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_1802_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_ceiling(A,X))) ) ).

% floor_le_ceiling
tff(fact_1803_exp__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)) = aa(A,A,inverse_inverse(A),aa(A,A,exp(A),X)) ) ).

% exp_minus
tff(fact_1804_powr__minus,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A3: A] : aa(A,A,aa(A,fun(A,A),powr(A),X),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),powr(A),X),A3)) ) ).

% powr_minus
tff(fact_1805_cosh__real__pos,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,cosh(real),X))) ).

% cosh_real_pos
tff(fact_1806_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(real,real,arcosh(real),aa(real,real,cosh(real),X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_1807_cosh__real__nonneg,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,cosh(real),X))) ).

% cosh_real_nonneg
tff(fact_1808_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,cosh(real),X)),aa(real,real,cosh(real),Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_1809_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),zero_zero(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,cosh(real),X)),aa(real,real,cosh(real),Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),X)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_1810_divide__real__def,axiom,
    ! [X: real,Y2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y2) = aa(real,real,aa(real,fun(real,real),times_times(real),X),aa(real,real,inverse_inverse(real),Y2)) ).

% divide_real_def
tff(fact_1811_cosh__real__ge__1,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,cosh(real),X))) ).

% cosh_real_ge_1
tff(fact_1812_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_1813_mult__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),K))) ) ) ).

% mult_less_mono1
tff(fact_1814_mult__less__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J2))) ) ) ).

% mult_less_mono2
tff(fact_1815_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_1816_mult__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) ).

% mult_Suc
tff(fact_1817_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) )
     => ( ( N = one_one(nat) )
        | ( M2 = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1818_less__mult__imp__div__less,axiom,
    ! [M2: nat,I: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_1819_div__times__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),N)),M2)) ).

% div_times_less_eq_dividend
tff(fact_1820_times__div__less__eq__dividend,axiom,
    ! [N: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N))),M2)) ).

% times_div_less_eq_dividend
tff(fact_1821_mod__eq__0D,axiom,
    ! [M2: nat,D2: nat] :
      ( ( modulo_modulo(nat,M2,D2) = zero_zero(nat) )
     => ? [Q3: nat] : M2 = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q3) ) ).

% mod_eq_0D
tff(fact_1822_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_1823_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% inverse_le_imp_le
tff(fact_1824_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% le_imp_inverse_le
tff(fact_1825_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_1826_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1827_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% inverse_le_1_iff
tff(fact_1828_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% one_less_inverse
tff(fact_1829_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% one_less_inverse_iff
tff(fact_1830_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_1831_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_1832_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_1833_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_1834_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_1835_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),X)) ) ) ).

% le_floor_iff
tff(fact_1836_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% floor_less_iff
tff(fact_1837_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_1838_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)))) ) ).

% le_floor_add
tff(fact_1839_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,X)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),X)) ) ).

% int_add_floor
tff(fact_1840_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_1841_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% sgn_1_pos
tff(fact_1842_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_1843_ceiling__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),X)) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,X)) ) ).

% ceiling_minus
tff(fact_1844_floor__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),X)) = aa(int,int,uminus_uminus(int),archimedean_ceiling(A,X)) ) ).

% floor_minus
tff(fact_1845_ceiling__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,X) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),X))) ) ).

% ceiling_def
tff(fact_1846_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(N),X)) = aa(real,real,sgn_sgn(real),X) ) ) ).

% sgn_root
tff(fact_1847_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: nat] :
          ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
         => ( archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),archim6421214686448440834_floor(A,X)),N) ) ) ) ).

% floor_power
tff(fact_1848_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( ( A3 = zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = zero_zero(A) ) )
          & ( ( A3 != zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ) ).

% abs_sgn_eq
tff(fact_1849_cosh__real__strict__mono,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cosh(real),X)),aa(real,real,cosh(real),Y2))) ) ) ).

% cosh_real_strict_mono
tff(fact_1850_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cosh(real),X)),aa(real,real,cosh(real),Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_1851_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),zero_zero(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cosh(real),X)),aa(real,real,cosh(real),Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_1852_inverse__powr,axiom,
    ! [Y2: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
     => ( aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,inverse_inverse(real),Y2)),A3) = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),powr(real),Y2),A3)) ) ) ).

% inverse_powr
tff(fact_1853_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N))) ) ) ).

% one_less_mult
tff(fact_1854_n__less__m__mult__n,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N))) ) ) ).

% n_less_m_mult_n
tff(fact_1855_n__less__n__mult__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2))) ) ) ).

% n_less_n_mult_m
tff(fact_1856_div__less__iff__less__mult,axiom,
    ! [Q5: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),Q5)),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5))) ) ) ).

% div_less_iff_less_mult
tff(fact_1857_mod__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q5: nat] : modulo_modulo(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N),Q5))),modulo_modulo(nat,M2,N)) ).

% mod_mult2_eq
tff(fact_1858_div__mod__decomp,axiom,
    ! [A4: nat,N: 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),N)),N)),modulo_modulo(nat,A4,N)) ).

% div_mod_decomp
tff(fact_1859_modulo__nat__def,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,M2,N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),N)) ).

% modulo_nat_def
tff(fact_1860_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_1861_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1862_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ).

% inverse_le_iff
tff(fact_1863_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ) ).

% inverse_less_iff
tff(fact_1864_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% one_le_inverse
tff(fact_1865_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% inverse_less_1_iff
tff(fact_1866_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ) ).

% one_le_inverse_iff
tff(fact_1867_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) ) ).

% one_add_floor
tff(fact_1868_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_1869_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),X)) ) ) ).

% reals_Archimedean
tff(fact_1870_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M2: nat,N: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)) ) ).

% floor_divide_of_nat_eq
tff(fact_1871_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_1872_sgn__real__def,axiom,
    ! [A3: real] :
      ( ( ( A3 = zero_zero(real) )
       => ( aa(real,real,sgn_sgn(real),A3) = zero_zero(real) ) )
      & ( ( A3 != zero_zero(real) )
       => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
           => ( aa(real,real,sgn_sgn(real),A3) = one_one(real) ) )
          & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
           => ( aa(real,real,sgn_sgn(real),A3) = aa(real,real,uminus_uminus(real),one_one(real)) ) ) ) ) ) ).

% sgn_real_def
tff(fact_1873_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = one_one(A) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).

% sgn_if
tff(fact_1874_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% sgn_1_neg
tff(fact_1875_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_1876_le__nat__floor,axiom,
    ! [X: nat,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),X)),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(int,nat,nat2,archim6421214686448440834_floor(real,A3)))) ) ).

% le_nat_floor
tff(fact_1877_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
           => ( archimedean_ceiling(A,X) = archim6421214686448440834_floor(A,X) ) )
          & ( ( X != aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ) ) ) ).

% ceiling_altdef
tff(fact_1878_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),archim6421214686448440834_floor(A,X))),one_one(int))) ) ).

% ceiling_diff_floor_le_1
tff(fact_1879_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(int,real,ring_1_of_int(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))))
       => ( archim6421214686448440834_floor(real,X) = N ) ) ) ).

% floor_eq
tff(fact_1880_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : pp(aa(real,bool,aa(real,fun(real,bool),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_1881_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : pp(aa(real,bool,aa(real,fun(real,bool),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_1882_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : pp(aa(real,bool,aa(real,fun(real,bool),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_1883_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : pp(aa(real,bool,aa(real,fun(real,bool),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_1884_forall__pos__mono__1,axiom,
    ! [P: fun(real,bool),E2: real] :
      ( ! [D5: real,E: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D5),E))
         => ( pp(aa(real,bool,P,D5))
           => pp(aa(real,bool,P,E)) ) )
     => ( ! [N2: nat] : pp(aa(real,bool,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
         => pp(aa(real,bool,P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_1885_forall__pos__mono,axiom,
    ! [P: fun(real,bool),E2: real] :
      ( ! [D5: real,E: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D5),E))
         => ( pp(aa(real,bool,P,D5))
           => pp(aa(real,bool,P,E)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero(nat) )
           => pp(aa(real,bool,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N2)))) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
         => pp(aa(real,bool,P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_1886_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
    <=> ? [N5: nat] :
          ( ( N5 != zero_zero(nat) )
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5))))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5))),E2)) ) ) ).

% real_arch_inverse
tff(fact_1887_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_inverse
tff(fact_1888_div__nat__eqI,axiom,
    ! [N: nat,Q5: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q5))))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = Q5 ) ) ) ).

% div_nat_eqI
tff(fact_1889_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q5: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Q5)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q5)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1890_split__div,axiom,
    ! [P: fun(nat,bool),M2: nat,N: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ( ( N != zero_zero(nat) )
         => ! [I2: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I2)),J3) )
               => pp(aa(nat,bool,P,I2)) ) ) ) ) ) ).

% split_div
tff(fact_1891_dividend__less__div__times,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_1892_dividend__less__times__div,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N))))) ) ).

% dividend_less_times_div
tff(fact_1893_mult__eq__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = zero_zero(nat) ) )
      & ( ( M2 != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))),N)) ) ) ) ).

% mult_eq_if
tff(fact_1894_split__mod,axiom,
    ! [P: fun(nat,bool),M2: nat,N: nat] :
      ( pp(aa(nat,bool,P,modulo_modulo(nat,M2,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,M2)) )
        & ( ( N != zero_zero(nat) )
         => ! [I2: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I2)),J3) )
               => pp(aa(nat,bool,P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_1895_nat__mult__distrib,axiom,
    ! [Z: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) ) ) ).

% nat_mult_distrib
tff(fact_1896_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))))
           => ( archim6421214686448440834_floor(A,X) = Z ) ) ) ) ).

% floor_unique
tff(fact_1897_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,X) = A3 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A)))) ) ) ) ).

% floor_eq_iff
tff(fact_1898_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T2)))
        <=> ! [I2: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I2)),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A)))) )
             => pp(aa(int,bool,P,I2)) ) ) ) ).

% floor_split
tff(fact_1899_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),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_1900_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X)) ) ) ).

% less_floor_iff
tff(fact_1901_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))) ) ) ).

% floor_le_iff
tff(fact_1902_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))))) ) ) ).

% floor_correct
tff(fact_1903_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N2))),X)) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_1904_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),M2)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_1905_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_1906_sgn__power__injE,axiom,
    ! [A3: real,N: nat,X: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A3)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A3)),N)) = X )
     => ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),N)) )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( A3 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_1907_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(int,real,ring_1_of_int(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))))
       => ( archim6421214686448440834_floor(real,X) = N ) ) ) ).

% floor_eq2
tff(fact_1908_floor__divide__real__eq__div,axiom,
    ! [B2: int,A3: real] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_1909_log__inverse,axiom,
    ! [A3: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,log2(A3),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,log2(A3),X)) ) ) ) ) ).

% log_inverse
tff(fact_1910_split__div_H,axiom,
    ! [P: fun(nat,bool),M2: nat,N: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)))
    <=> ( ( ( N = zero_zero(nat) )
          & pp(aa(nat,bool,P,zero_zero(nat))) )
        | ? [Q4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)),M2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q4))))
            & pp(aa(nat,bool,P,Q4)) ) ) ) ).

% split_div'
tff(fact_1911_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),M2) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1912_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q5)))),Q5)),P2)) ) ) ).

% floor_divide_lower
tff(fact_1913_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z2)) = 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),Z2))) ) ) ).

% nat_mult_distrib_neg
tff(fact_1914_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),aa(real,real,root(N),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),aa(real,real,root(N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_1915_root__sgn__power,axiom,
    ! [N: nat,Y2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y2)),N))) = Y2 ) ) ).

% root_sgn_power
tff(fact_1916_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q5)))),one_one(A))),Q5))) ) ) ).

% floor_divide_upper
tff(fact_1917_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1918_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ( K = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = zero_zero(nat) ) )
      & ( ( K != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_1919_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1920_sgn__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_one
tff(fact_1921_nat__less__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),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),J2),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_1922_nat__less__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J2)),U)),M2)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_1923_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_1924_nat__diff__add__eq2,axiom,
    ! [I: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( 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)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),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),J2),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1925_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L2: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L2)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_sgnE
tff(fact_1926_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_eq_sgn_abs
tff(fact_1927_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero(int) )
       => ( aa(int,int,sgn_sgn(int),I) = zero_zero(int) ) )
      & ( ( I != zero_zero(int) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I))
           => ( aa(int,int,sgn_sgn(int),I) = one_one(int) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I))
           => ( aa(int,int,sgn_sgn(int),I) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ) ).

% zsgn_def
tff(fact_1928_div__sgn__abs__cancel,axiom,
    ! [V2: int,K: int,L: int] :
      ( ( V2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V2)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V2)),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_1929_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( K = zero_zero(nat) )
        | ( M2 = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1930_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) )
        <=> ( X = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_1931_Real__Vector__Spaces_Osgn__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),X)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),X)) ) ).

% Real_Vector_Spaces.sgn_minus
tff(fact_1932_nat__mult__less__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_1933_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
      <=> ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1934_real__sgn__eq,axiom,
    ! [X: real] : aa(real,real,sgn_sgn(real),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,abs_abs(real),X)) ).

% real_sgn_eq
tff(fact_1935_nat__mult__le__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_1936_nat__mult__div__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_1937_nat__eq__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),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)),M2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J2)),U)),M2) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1938_nat__eq__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N) )
      <=> ( M2 = 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),J2),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1939_nat__le__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J2)),U)),M2)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_1940_nat__le__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),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),J2),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_1941_nat__diff__add__eq1,axiom,
    ! [J2: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),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)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)) = 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),J2)),U)),M2)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1942_Cauchy__iff2,axiom,
    ! [X6: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X6)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),M5))
         => ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N5))
             => pp(aa(real,bool,aa(real,fun(real,bool),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,N5)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_1943_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y2))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y2))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y2))),one_one(int)) ) ) ) ) ).

% floor_add
tff(fact_1944_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_1945_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_1946_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_1947_mult__ceiling__le__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A3: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A3)),archimedean_ceiling(B,B2))))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_1948_le__mult__floor__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A3: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A3)),archim6421214686448440834_floor(B,B2)))),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2))))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_1949_frac__frac,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,archimedean_frac(A,X)) = archimedean_frac(A,X) ) ).

% frac_frac
tff(fact_1950_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_1951_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,one_one(nat)) = A3 ) ).

% pochhammer_1
tff(fact_1952_frac__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),archimedean_frac(A,X)),ring_1_Ints(A)))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_in_Ints_iff
tff(fact_1953_gbinomial__0_I2_J,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [K: nat] : aa(nat,B,gbinomial(B,zero_zero(B)),aa(nat,nat,suc,K)) = zero_zero(B) ) ).

% gbinomial_0(2)
tff(fact_1954_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_1955_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_1956_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% gbinomial_Suc0
tff(fact_1957_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% pochhammer_Suc0
tff(fact_1958_frac__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archimedean_frac(A,aa(int,A,ring_1_of_int(A),Z)) = zero_zero(A) ) ).

% frac_of_int
tff(fact_1959_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_eq_0_iff
tff(fact_1960_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
            | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),ring_1_Ints(A))) )
         => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y2)) ) ) ) ).

% floor_add2
tff(fact_1961_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),archimedean_frac(A,X)))
        <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_gt_0_iff
tff(fact_1962_of__nat__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,gbinomial(nat,N),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) ) ).

% of_nat_gbinomial
tff(fact_1963_pochhammer__of__nat,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: nat,N: nat] : comm_s3205402744901411588hammer(A,aa(nat,A,semiring_1_of_nat(A),X),N) = aa(nat,A,semiring_1_of_nat(A),comm_s3205402744901411588hammer(nat,X,N)) ) ).

% pochhammer_of_nat
tff(fact_1964_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),ring_1_Ints(A))) ) ).

% Ints_0
tff(fact_1965_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_mult
tff(fact_1966_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),ring_1_Ints(A))) ) ).

% Ints_1
tff(fact_1967_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_add
tff(fact_1968_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_diff
tff(fact_1969_minus__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,uminus_uminus(A),X)),ring_1_Ints(A)))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% minus_in_Ints_iff
tff(fact_1970_Ints__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,uminus_uminus(A),A3)),ring_1_Ints(A))) ) ) ).

% Ints_minus
tff(fact_1971_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),ring_1_Ints(A))) ) ) ).

% Ints_power
tff(fact_1972_Ints__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,semiring_1_of_nat(A),N)),ring_1_Ints(A))) ) ).

% Ints_of_nat
tff(fact_1973_Ints__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,abs_abs(A),A3)),ring_1_Ints(A))) ) ) ).

% Ints_abs
tff(fact_1974_Ints__of__int,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(int,A,ring_1_of_int(A),Z)),ring_1_Ints(A))) ) ).

% Ints_of_int
tff(fact_1975_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q5: A,P: fun(A,bool)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Q5),ring_1_Ints(A)))
         => ( ! [Z3: int] : pp(aa(A,bool,P,aa(int,A,ring_1_of_int(A),Z3)))
           => pp(aa(A,bool,P,Q5)) ) ) ) ).

% Ints_induct
tff(fact_1976_Ints__cases,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Q5),ring_1_Ints(A)))
         => ~ ! [Z3: int] : Q5 != aa(int,A,ring_1_of_int(A),Z3) ) ) ).

% Ints_cases
tff(fact_1977_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_1978_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_1979_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_pos
tff(fact_1980_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),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_1981_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,M2) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( comm_s3205402744901411588hammer(A,A3,N) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_1982_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat,M2: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( comm_s3205402744901411588hammer(A,A3,M2) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_1983_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = zero_zero(A) ) )
          & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,X)) ) ) ) ) ).

% frac_neg
tff(fact_1984_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,X))) ) ).

% frac_ge_0
tff(fact_1985_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),archimedean_frac(A,X)),one_one(A))) ) ).

% frac_lt_1
tff(fact_1986_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = archimedean_frac(A,X) ) ).

% frac_1_eq
tff(fact_1987_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: A] :
          ( ( archimedean_frac(A,X) = A3 )
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3)),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) ) ) ) ).

% frac_unique_iff
tff(fact_1988_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_1989_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_1990_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),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_1991_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_1992_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_1993_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_nonneg
tff(fact_1994_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = zero_zero(A) ) ) ) ) ).

% pochhammer_0_left
tff(fact_1995_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),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_1996_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_1997_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_1998_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M2)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M2)),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),M2),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_1999_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% Ints_odd_less_0
tff(fact_2000_frac__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))) ) ).

% frac_def
tff(fact_2001_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = 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)),N)) ) ).

% pochhammer_rec
tff(fact_2002_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A3,N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% pochhammer_Suc
tff(fact_2003_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,N: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),N))),comm_s3205402744901411588hammer(A,Z,N)) ) ).

% pochhammer_rec'
tff(fact_2004_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),N))
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2005_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2006_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2007_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( ( X != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_2008_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_2009_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2010_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,N: nat,M2: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),N)),M2)) ) ).

% pochhammer_product'
tff(fact_2011_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),ring_1_Ints(A)))
           => ( ( X = Y2 )
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2))),one_one(A))) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_2012_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_2013_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_2014_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_2015_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,gbinomial(A,aa(A,A,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))),N)) ) ).

% gbinomial_index_swap
tff(fact_2016_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_2017_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% frac_eq
tff(fact_2018_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_2019_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y2))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y2))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = 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,X)),archimedean_frac(A,Y2))),one_one(A)) ) ) ) ) ).

% frac_add
tff(fact_2020_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M2: nat,N: nat,Z: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( comm_s3205402744901411588hammer(A,Z,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M2)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% pochhammer_product
tff(fact_2021_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_2022_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_2023_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_2024_power_Opower__eq__if,axiom,
    ! [A: $tType,M2: nat,One: A,Times: fun(A,fun(A,A)),P2: A] :
      ( ( ( M2 = zero_zero(nat) )
       => ( power2(A,One,Times,P2,M2) = One ) )
      & ( ( M2 != zero_zero(nat) )
       => ( power2(A,One,Times,P2,M2) = aa(A,A,aa(A,fun(A,A),Times,P2),power2(A,One,Times,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat)))) ) ) ) ).

% power.power_eq_if
tff(fact_2025_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2026_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),semiring_char_0_fact(A,N)) ) ).

% pochhammer_same
tff(fact_2027_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_2028_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M2: nat] :
          ( ( ( M2 = zero_zero(nat) )
           => ( semiring_char_0_fact(A,M2) = one_one(A) ) )
          & ( ( M2 != zero_zero(nat) )
           => ( semiring_char_0_fact(A,M2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat)))) ) ) ) ) ).

% fact_num_eq_if
tff(fact_2029_of__nat__fact,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),semiring_char_0_fact(nat,N)) = semiring_char_0_fact(A,N) ) ).

% of_nat_fact
tff(fact_2030_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_2031_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_2032_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_2033_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_2034_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N))),semiring_char_0_fact(A,N)) ) ).

% fact_Suc
tff(fact_2035_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_2036_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),semiring_char_0_fact(nat,M2)),semiring_char_0_fact(nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_2037_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_zero
tff(fact_2038_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,N)),zero_zero(A))) ) ).

% fact_not_neg
tff(fact_2039_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_gt_zero
tff(fact_2040_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_1
tff(fact_2041_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) = comm_s3205402744901411588hammer(A,one_one(A),N) ) ).

% pochhammer_fact
tff(fact_2042_power_Opower_Opower__Suc,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A,N: nat] : power2(A,One,Times,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),Times,A3),power2(A,One,Times,A3,N)) ).

% power.power.power_Suc
tff(fact_2043_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A] : power2(A,One,Times,A3,zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_2044_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_2045_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,M2)),semiring_char_0_fact(A,N))) ) ) ) ).

% fact_less_mono
tff(fact_2046_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( modulo_modulo(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,M2)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_2047_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,N)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),N)))) ) ).

% fact_le_power
tff(fact_2048_fact__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2)))
     => ( semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),N)) = 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,M2)),N)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))) ) ) ).

% fact_diff_Suc
tff(fact_2049_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),R)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R))) ) ).

% fact_div_fact_le_pow
tff(fact_2050_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2051_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2052_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M2: nat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),one_one(real)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,W)),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),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),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W),M2)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M2)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W))))) ) ) ) ).

% norm_power_diff
tff(fact_2053_cppi,axiom,
    ! [D6: int,P: fun(int,bool),P3: fun(int,bool),A4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),X2))
           => ( pp(aa(int,bool,P,X2))
            <=> pp(aa(int,bool,P3,X2)) ) )
       => ( ! [X2: int] :
              ( ! [Xa2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                 => ! [Xb: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A4))
                     => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
             => ( pp(aa(int,bool,P,X2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6))) ) )
         => ( ! [X2: int,K2: int] :
                ( pp(aa(int,bool,P3,X2))
              <=> pp(aa(int,bool,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D6)))) )
           => ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                    & pp(aa(int,bool,P3,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                    & ? [Xa3: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),A4))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2054_cpmi,axiom,
    ! [D6: int,P: fun(int,bool),P3: fun(int,bool),B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),Z4))
           => ( pp(aa(int,bool,P,X2))
            <=> pp(aa(int,bool,P3,X2)) ) )
       => ( ! [X2: int] :
              ( ! [Xa2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                 => ! [Xb: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B5))
                     => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
             => ( pp(aa(int,bool,P,X2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6))) ) )
         => ( ! [X2: int,K2: int] :
                ( pp(aa(int,bool,P3,X2))
              <=> pp(aa(int,bool,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D6)))) )
           => ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                    & pp(aa(int,bool,P3,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                    & ? [Xa3: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),B5))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2055_bset_I6_J,axiom,
    ! [D6: int,B5: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X3),T2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6)),T2)) ) ) ) ).

% bset(6)
tff(fact_2056_bset_I8_J,axiom,
    ! [D6: int,T2: int,B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int))),B5))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X3))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6))) ) ) ) ) ).

% bset(8)
tff(fact_2057_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M2: num,N: num] :
          ( ( aa(num,A,numeral_numeral(A),M2) = aa(num,A,numeral_numeral(A),N) )
        <=> ( M2 = N ) ) ) ).

% numeral_eq_iff
tff(fact_2058_int__eq__iff__numeral,axiom,
    ! [M2: nat,V2: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M2) = aa(num,int,numeral_numeral(int),V2) )
    <=> ( M2 = aa(num,nat,numeral_numeral(nat),V2) ) ) ).

% int_eq_iff_numeral
tff(fact_2059_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_2060_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N)) ) ) ).

% numeral_le_iff
tff(fact_2061_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N)) ) ) ).

% numeral_less_iff
tff(fact_2062_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V2: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),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),V2),W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_2063_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ) ).

% numeral_times_numeral
tff(fact_2064_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V2: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),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),V2),W))),Z) ) ).

% add_numeral_left
tff(fact_2065_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)) ) ).

% numeral_plus_numeral
tff(fact_2066_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_2067_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M2: num,N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( M2 = N ) ) ) ).

% neg_numeral_eq_iff
tff(fact_2068_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% of_nat_numeral
tff(fact_2069_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_numeral
tff(fact_2070_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_2071_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,N: num] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = aa(num,A,numeral_numeral(A),N) )
        <=> ( Z = aa(num,int,numeral_numeral(int),N) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_2072_norm__minus__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V7770717601297561774m_norm(A,aa(A,A,uminus_uminus(A),X)) = real_V7770717601297561774m_norm(A,X) ) ).

% norm_minus_cancel
tff(fact_2073_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V2)) = aa(num,int,numeral_numeral(int),V2) ) ).

% floor_numeral
tff(fact_2074_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V2)) = aa(num,int,numeral_numeral(int),V2) ) ).

% ceiling_numeral
tff(fact_2075_norm__fact,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [N: nat] : real_V7770717601297561774m_norm(A,semiring_char_0_fact(A,N)) = semiring_char_0_fact(real,N) ) ).

% norm_fact
tff(fact_2076_numeral__powr__numeral__real,axiom,
    ! [M2: num,N: num] : aa(real,real,aa(real,fun(real,real),powr(real),aa(num,real,numeral_numeral(real),M2)),aa(num,real,numeral_numeral(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),M2)),aa(num,nat,numeral_numeral(nat),N)) ).

% numeral_powr_numeral_real
tff(fact_2077_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M2)) ) ) ).

% neg_numeral_le_iff
tff(fact_2078_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B2: A,V2: 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),V2)) = 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),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ).

% distrib_right_numeral
tff(fact_2079_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),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),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).

% distrib_left_numeral
tff(fact_2080_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M2)) ) ) ).

% neg_numeral_less_iff
tff(fact_2081_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A3: A,B2: A,V2: 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),V2)) = 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),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ).

% left_diff_distrib_numeral
tff(fact_2082_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),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),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_2083_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_2084_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_2085_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_2086_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W: num,Y2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y2)) = 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),V2),W)))),Y2) ) ).

% semiring_norm(170)
tff(fact_2087_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W: num,Y2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y2)) = 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),V2),W)))),Y2) ) ).

% semiring_norm(171)
tff(fact_2088_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W: num,Y2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y2)) = 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),V2),W))),Y2) ) ).

% semiring_norm(172)
tff(fact_2089_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_2090_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W: num,Y2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y2)) = 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),V2),W)))),Y2) ) ).

% semiring_norm(168)
tff(fact_2091_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ).

% diff_numeral_simps(3)
tff(fact_2092_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)) ) ).

% diff_numeral_simps(2)
tff(fact_2093_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_neg_numeral
tff(fact_2094_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2095_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( real_V7770717601297561774m_norm(A,X) = zero_zero(real) )
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2096_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_2097_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_2098_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(num,real,numeral_numeral(real),W)),aa(nat,real,semiring_1_of_nat(real),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),W)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_2099_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(num,real,numeral_numeral(real),W)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),W))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_2100_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M2: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),N)),aa(nat,real,semiring_1_of_nat(real),M2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),N)),M2)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_2101_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_2102_norm__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [N: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,real,semiring_1_of_nat(real),N) ) ).

% norm_of_nat
tff(fact_2103_diff__nat__numeral,axiom,
    ! [V2: num,V3: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),aa(num,nat,numeral_numeral(nat),V3)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),V3))) ).

% diff_nat_numeral
tff(fact_2104_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y2: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = aa(int,nat,nat2,Y2) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y2 ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_2105_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,N: nat] :
      ( ( aa(int,nat,nat2,Y2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) )
    <=> ( Y2 = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_2106_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_2107_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2108_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2109_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 )
        <=> ( ( ( 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)) ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2110_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)) )
        <=> ( ( ( 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 ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2111_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2112_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2113_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_2114_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,X)))
        <=> ( X != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2115_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real)))
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2116_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_le_iff
tff(fact_2117_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_le_numeral_iff
tff(fact_2118_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_less_iff
tff(fact_2119_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_less_numeral_iff
tff(fact_2120_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_2121_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: num,N: nat,Y2: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) = aa(nat,A,semiring_1_of_nat(A),Y2) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = Y2 ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_2122_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y2: nat,X: num,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Y2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) )
        <=> ( Y2 = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_2123_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),V2)),X)) ) ) ).

% numeral_le_floor
tff(fact_2124_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% floor_less_numeral
tff(fact_2125_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% ceiling_le_numeral
tff(fact_2126_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),V2)),X)) ) ) ).

% numeral_less_ceiling
tff(fact_2127_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ).

% floor_neg_numeral
tff(fact_2128_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_add_numeral
tff(fact_2129_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% floor_diff_numeral
tff(fact_2130_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_neg_numeral
tff(fact_2131_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y2: int,X: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) )
        <=> ( Y2 = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_2132_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y2: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) = aa(int,A,ring_1_of_int(A),Y2) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y2 ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_2133_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_diff_numeral
tff(fact_2134_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),powr(real),X),aa(num,real,numeral_numeral(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),N)) ) ) ).

% powr_numeral
tff(fact_2135_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),N)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2136_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,N: nat] : archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ).

% floor_numeral_power
tff(fact_2137_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,N: nat] : archimedean_ceiling(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ).

% ceiling_numeral_power
tff(fact_2138_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_2139_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2140_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2141_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 )
        <=> ( ( ( 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))) ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2142_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))) )
        <=> ( ( ( 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 ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2143_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2144_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2145_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_2146_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_2147_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_2148_nat__numeral__diff__1,axiom,
    ! [V2: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V2)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_2149_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A3: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A3)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2150_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2151_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A3: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A3)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2152_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_2153_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_2154_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_2155_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_2156_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,N: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X)) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2157_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N))) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2158_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,N: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X)) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2159_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N))) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2160_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).

% numeral_less_floor
tff(fact_2161_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).

% floor_le_numeral
tff(fact_2162_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).

% ceiling_less_numeral
tff(fact_2163_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).

% numeral_le_ceiling
tff(fact_2164_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X)) ) ) ).

% neg_numeral_le_floor
tff(fact_2165_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)))) ) ) ).

% floor_less_neg_numeral
tff(fact_2166_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2167_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2168_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2169_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X)) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2170_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2171_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2172_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y2: int,X: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) )
        <=> ( Y2 = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_2173_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y2: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) = aa(int,A,ring_1_of_int(A),Y2) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) = Y2 ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_2174_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_2175_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X)) ) ) ).

% neg_numeral_less_floor
tff(fact_2176_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A)))) ) ) ).

% floor_le_neg_numeral
tff(fact_2177_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A)))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2178_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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),V2))),one_one(A))),X)) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2179_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),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),X))),N))) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2180_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3)) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2181_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),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),X))),N))) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2182_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3)) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2183_int__ops_I3_J,axiom,
    ! [N: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),N)) = aa(num,int,numeral_numeral(int),N) ).

% int_ops(3)
tff(fact_2184_nat__numeral__as__int,axiom,
    ! [X3: num] : aa(num,nat,numeral_numeral(nat),X3) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X3)) ).

% nat_numeral_as_int
tff(fact_2185_norm__minus__commute,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ).

% norm_minus_commute
tff(fact_2186_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),N) ) ).

% zero_neq_numeral
tff(fact_2187_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_2188_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M2: num,N: num] : aa(num,A,numeral_numeral(A),M2) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_neq_neg_numeral
tff(fact_2189_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M2: num,N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)) != aa(num,A,numeral_numeral(A),N) ) ).

% neg_numeral_neq_numeral
tff(fact_2190_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: num] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(num,A,numeral_numeral(A),N)),ring_1_Ints(A))) ) ).

% Ints_numeral
tff(fact_2191_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_2192_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
              | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
                & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3))
                  | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2)) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2193_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_2194_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_le_numeral
tff(fact_2195_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_le_zero
tff(fact_2196_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_less_numeral
tff(fact_2197_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_less_zero
tff(fact_2198_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% one_le_numeral
tff(fact_2199_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A))) ) ).

% not_numeral_less_one
tff(fact_2200_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_le_neg_numeral
tff(fact_2201_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_le_numeral
tff(fact_2202_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% zero_neq_neg_numeral
tff(fact_2203_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_less_numeral
tff(fact_2204_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2205_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_2206_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% numeral_times_minus_swap
tff(fact_2207_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),N) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_2208_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% one_neq_neg_numeral
tff(fact_2209_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),X)),Y2)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) ) ).

% norm_uminus_minus
tff(fact_2210_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_2211_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,N: nat,Z: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( real_V7770717601297561774m_norm(A,W) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2212_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y2: A,E1: real,Z: A,E22: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2))),E1))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y2),Z))),E22))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2213_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y2)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2))))) ) ).

% norm_triangle_sub
tff(fact_2214_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B2)))) ) ).

% norm_triangle_ineq4
tff(fact_2215_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y2: A,E1: real,Z: A,E22: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2))),E1))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y2),Z))),E22))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% norm_diff_triangle_le
tff(fact_2216_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y2: A,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y2))),E2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2))),E2)) ) ) ).

% norm_triangle_le_diff
tff(fact_2217_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_le_zero
tff(fact_2218_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_le_neg_numeral
tff(fact_2219_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_less_zero
tff(fact_2220_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_less_neg_numeral
tff(fact_2221_bset_I1_J,axiom,
    ! [D6: int,B5: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X2: int] :
          ( ! [Xa2: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B5))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
         => ( pp(aa(int,bool,P,X2))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6))) ) )
     => ( ! [X2: int] :
            ( ! [Xa2: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B5))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
           => ( pp(aa(int,bool,Q,X2))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6))) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( ( pp(aa(int,bool,P,X3))
                & pp(aa(int,bool,Q,X3)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6)))
                & pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6))) ) ) ) ) ) ).

% bset(1)
tff(fact_2222_bset_I2_J,axiom,
    ! [D6: int,B5: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X2: int] :
          ( ! [Xa2: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B5))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
         => ( pp(aa(int,bool,P,X2))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6))) ) )
     => ( ! [X2: int] :
            ( ! [Xa2: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B5))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
           => ( pp(aa(int,bool,Q,X2))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6))) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( ( pp(aa(int,bool,P,X3))
                | pp(aa(int,bool,Q,X3)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6)))
                | pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6))) ) ) ) ) ) ).

% bset(2)
tff(fact_2223_aset_I1_J,axiom,
    ! [D6: int,A4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X2: int] :
          ( ! [Xa2: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A4))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
         => ( pp(aa(int,bool,P,X2))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6))) ) )
     => ( ! [X2: int] :
            ( ! [Xa2: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A4))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
           => ( pp(aa(int,bool,Q,X2))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6))) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
           => ( ( pp(aa(int,bool,P,X3))
                & pp(aa(int,bool,Q,X3)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)))
                & pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6))) ) ) ) ) ) ).

% aset(1)
tff(fact_2224_aset_I2_J,axiom,
    ! [D6: int,A4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X2: int] :
          ( ! [Xa2: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A4))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
         => ( pp(aa(int,bool,P,X2))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6))) ) )
     => ( ! [X2: int] :
            ( ! [Xa2: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A4))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
           => ( pp(aa(int,bool,Q,X2))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6))) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
           => ( ( pp(aa(int,bool,P,X3))
                | pp(aa(int,bool,Q,X3)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)))
                | pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6))) ) ) ) ) ) ).

% aset(2)
tff(fact_2225_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A] : pp(aa(real,bool,aa(real,fun(real,bool),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_2226_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2227_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2228_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A] : pp(aa(real,bool,aa(real,fun(real,bool),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),minus_minus(A),A3),B2)))) ) ).

% norm_triangle_ineq2
tff(fact_2229_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A))) ) ).

% neg_numeral_le_one
tff(fact_2230_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M2))) ) ).

% neg_one_le_numeral
tff(fact_2231_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% neg_numeral_le_neg_one
tff(fact_2232_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_le_neg_one
tff(fact_2233_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))) ) ).

% not_one_le_neg_numeral
tff(fact_2234_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A))) ) ).

% neg_numeral_less_one
tff(fact_2235_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M2))) ) ).

% neg_one_less_numeral
tff(fact_2236_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_less_neg_one
tff(fact_2237_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))) ) ).

% not_one_less_neg_numeral
tff(fact_2238_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2239_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A3)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A3)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_2240_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),X))),aa(real,real,exp(real),real_V7770717601297561774m_norm(A,X)))) ) ).

% norm_exp
tff(fact_2241_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),powr(real),X),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),N))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),N))) ) ) ).

% powr_neg_numeral
tff(fact_2242_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),N) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W) = one_one(real) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2243_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A,C2: A,D2: A] : pp(aa(real,bool,aa(real,fun(real,bool),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_2244_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),X)) = zero_zero(real) ) )
          & ( ( X != zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),X)) = one_one(real) ) ) ) ) ).

% norm_sgn
tff(fact_2245_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2246_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2247_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2248_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2249_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B2)))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))) ) ).

% norm_triangle_ineq3
tff(fact_2250_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X2: int,K2: int] :
            ( pp(aa(int,bool,P,X2))
          <=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
        <=> ? [X4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D2)))
              & pp(aa(int,bool,P,X4)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2251_bset_I3_J,axiom,
    ! [D6: int,T2: int,B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int))),B5))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( ( X3 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6) = T2 ) ) ) ) ) ).

% bset(3)
tff(fact_2252_bset_I4_J,axiom,
    ! [D6: int,T2: int,B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),B5))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( ( X3 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6) != T2 ) ) ) ) ) ).

% bset(4)
tff(fact_2253_bset_I5_J,axiom,
    ! [D6: int,B5: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),T2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6)),T2)) ) ) ) ).

% bset(5)
tff(fact_2254_bset_I7_J,axiom,
    ! [D6: int,T2: int,B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),B5))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X3))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6))) ) ) ) ) ).

% bset(7)
tff(fact_2255_aset_I3_J,axiom,
    ! [D6: int,T2: int,A4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
           => ( ( X3 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6) = T2 ) ) ) ) ) ).

% aset(3)
tff(fact_2256_aset_I4_J,axiom,
    ! [D6: int,T2: int,A4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),A4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
           => ( ( X3 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6) != T2 ) ) ) ) ) ).

% aset(4)
tff(fact_2257_aset_I5_J,axiom,
    ! [D6: int,T2: int,A4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),A4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),T2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)),T2)) ) ) ) ) ).

% aset(5)
tff(fact_2258_aset_I7_J,axiom,
    ! [D6: int,A4: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X3))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6))) ) ) ) ).

% aset(7)
tff(fact_2259_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K5))
              & ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),K5)) )
        <=> ? [N7: nat] :
            ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% lemma_NBseq_def
tff(fact_2260_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K5))
              & ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),K5)) )
        <=> ? [N7: nat] :
            ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% lemma_NBseq_def2
tff(fact_2261_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2262_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2263_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2264_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2265_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [M7: nat] :
                ! [M5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),M5))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),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,N5)))),E3)) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_2266_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M8: nat] :
                ! [M3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M3))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,M3)),aa(nat,A,X6,N2)))),E)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI
tff(fact_2267_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
           => ? [M9: nat] :
              ! [M: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M))
               => ! [N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N4))
                   => pp(aa(real,bool,aa(real,fun(real,bool),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,N4)))),E2)) ) ) ) ) ) ).

% CauchyD
tff(fact_2268_aset_I8_J,axiom,
    ! [D6: int,A4: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X3))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6))) ) ) ) ).

% aset(8)
tff(fact_2269_aset_I6_J,axiom,
    ! [D6: int,T2: int,A4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X3),T2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)),T2)) ) ) ) ) ).

% aset(6)
tff(fact_2270_enat__ord__number_I2_J,axiom,
    ! [M2: num,N: num] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M2)),aa(num,extended_enat,numeral_numeral(extended_enat),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(2)
tff(fact_2271_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K6: real,N: nat] :
          ( ( H != zero_zero(A) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K6))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K6))
             => pp(aa(real,bool,aa(real,fun(real,bool),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)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K6),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),real_V7770717601297561774m_norm(A,H)))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2272_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A3: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),B2),A3))),real_V7770717601297561774m_norm(complex,B2))),real_V7770717601297561774m_norm(complex,A3))) ).

% complex_mod_triangle_ineq2
tff(fact_2273_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,X))),real_V7770717601297561774m_norm(complex,X))) ).

% complex_mod_minus_le_complex_mod
tff(fact_2274_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,X)),one_one(A))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_2275_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( ( archimedean_ceiling(real,aa(real,real,log2(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) )
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_2276_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: 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),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_2277_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y23: num] :
      ( ( aa(num,num,bit0,X23) = aa(num,num,bit0,Y23) )
    <=> ( X23 = Y23 ) ) ).

% verit_eq_simplify(8)
tff(fact_2278_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_2279_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(Tree),X))
     => ( vEBT_invar_vebt(Tree,N)
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N))) ) ) ).

% member_bound
tff(fact_2280_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_2281_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( aa(num,A,numeral_numeral(A),N) = one_one(A) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_2282_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),N) )
        <=> ( one2 = N ) ) ) ).

% one_eq_numeral_iff
tff(fact_2283_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( real_Vector_of_real(A,X) = zero_zero(A) )
        <=> ( X = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_2284_of__real__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( real_Vector_of_real(A,zero_zero(real)) = zero_zero(A) ) ) ).

% of_real_0
tff(fact_2285_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( real_Vector_of_real(A,X) = one_one(A) )
        <=> ( X = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_2286_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_2287_zdiv__numeral__Bit0,axiom,
    ! [V2: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,V2))),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),V2)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_2288_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [X: real,Y2: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,X)),real_Vector_of_real(A,Y2)) ) ).

% of_real_divide
tff(fact_2289_of__real__eq__minus__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y2: real] :
          ( ( real_Vector_of_real(A,X) = aa(A,A,uminus_uminus(A),real_Vector_of_real(A,Y2)) )
        <=> ( X = aa(real,real,uminus_uminus(real),Y2) ) ) ) ).

% of_real_eq_minus_of_real_iff
tff(fact_2290_minus__of__real__eq__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y2: real] :
          ( ( aa(A,A,uminus_uminus(A),real_Vector_of_real(A,X)) = real_Vector_of_real(A,Y2) )
        <=> ( aa(real,real,uminus_uminus(real),X) = Y2 ) ) ) ).

% minus_of_real_eq_of_real_iff
tff(fact_2291_of__real__minus,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] : real_Vector_of_real(A,aa(real,real,uminus_uminus(real),X)) = aa(A,A,uminus_uminus(A),real_Vector_of_real(A,X)) ) ).

% of_real_minus
tff(fact_2292_of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y2: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_Vector_of_real(A,X)),real_Vector_of_real(A,Y2)) ) ).

% of_real_diff
tff(fact_2293_of__real__of__nat__eq,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [N: nat] : real_Vector_of_real(A,aa(nat,real,semiring_1_of_nat(real),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% of_real_of_nat_eq
tff(fact_2294_num__double,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,one2)),N) = aa(num,num,bit0,N) ).

% num_double
tff(fact_2295_of__real__fact,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [N: nat] : real_Vector_of_real(A,semiring_char_0_fact(real,N)) = semiring_char_0_fact(A,N) ) ).

% of_real_fact
tff(fact_2296_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2297_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( N = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2298_Suc__numeral,axiom,
    ! [N: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),N)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% Suc_numeral
tff(fact_2299_inrange,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(T2)),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))),N)),one_one(nat))))) ) ).

% inrange
tff(fact_2300_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))))
        <=> ( M2 != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2301_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A))))
        <=> ( M2 != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2302_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_2303_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_2304_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_2305_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_2306_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_2307_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc'
tff(fact_2308_add__2__eq__Suc,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc
tff(fact_2309_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_2310_div2__Suc__Suc,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,M2))),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),M2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% div2_Suc_Suc
tff(fact_2311_add__self__div__2,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),M2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = M2 ).

% add_self_div_2
tff(fact_2312_mod2__Suc__Suc,axiom,
    ! [M2: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% mod2_Suc_Suc
tff(fact_2313_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% one_plus_numeral
tff(fact_2314_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ) ).

% numeral_plus_one
tff(fact_2315_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_2316_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),one2)) ) ) ).

% numeral_le_one_iff
tff(fact_2317_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),N)) ) ) ).

% one_less_numeral_iff
tff(fact_2318_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_2319_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M2))) ) ).

% add_neg_numeral_special(6)
tff(fact_2320_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M2)) ) ).

% diff_numeral_special(6)
tff(fact_2321_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_2322_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_2323_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2324_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
            <=> ( X = Y2 ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2325_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_2326_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2327_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_2328_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_2329_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_2330_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y2 = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2331_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_2332_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_2333_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_2334_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_2335_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: 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))),N)) = 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))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_2336_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2337_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),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),M2),one2))) ) ).

% diff_numeral_special(4)
tff(fact_2338_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: 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),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% diff_numeral_special(3)
tff(fact_2339_add__self__mod__2,axiom,
    ! [M2: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),M2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_2340_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% half_nonnegative_int_iff
tff(fact_2341_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% half_negative_int_iff
tff(fact_2342_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_2343_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_2344_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = one_one(A) ) ).

% power_minus1_even
tff(fact_2345_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) ) ) ).

% one_less_floor
tff(fact_2346_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ).

% floor_le_one
tff(fact_2347_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    <=> ( modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2348_square__powr__half,axiom,
    ! [X: real] : aa(real,real,aa(real,fun(real,real),powr(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),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),X) ).

% square_powr_half
tff(fact_2349_sgn__eq,axiom,
    ! [Z: complex] : aa(complex,complex,sgn_sgn(complex),Z) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),real_Vector_of_real(complex,real_V7770717601297561774m_norm(complex,Z))) ).

% sgn_eq
tff(fact_2350_mult__inc,axiom,
    ! [X: num,Y2: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X),inc(Y2)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),X),Y2)),X) ).

% mult_inc
tff(fact_2351_add__One,axiom,
    ! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2) = inc(X) ).

% add_One
tff(fact_2352_add__inc,axiom,
    ! [X: num,Y2: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),inc(Y2)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y2)) ).

% add_inc
tff(fact_2353_add__One__commute,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N) = aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2) ).

% add_One_commute
tff(fact_2354_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),X),one2))
    <=> ( X = one2 ) ) ).

% le_num_One_iff
tff(fact_2355_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one2 != aa(num,num,bit0,X23) ).

% verit_eq_simplify(10)
tff(fact_2356_inc_Osimps_I1_J,axiom,
    inc(one2) = aa(num,num,bit0,one2) ).

% inc.simps(1)
tff(fact_2357_num__induct,axiom,
    ! [P: fun(num,bool),X: num] :
      ( pp(aa(num,bool,P,one2))
     => ( ! [X2: num] :
            ( pp(aa(num,bool,P,X2))
           => pp(aa(num,bool,P,inc(X2))) )
       => pp(aa(num,bool,P,X)) ) ) ).

% num_induct
tff(fact_2358_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_2359_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),N) ) ).

% numeral_Bit0_div_2
tff(fact_2360_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_2361_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)),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),Y2),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_commute
tff(fact_2362_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [X: A,Y2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
        <=> ( ( X = Y2 )
            | ( X = aa(A,A,uminus_uminus(A),Y2) ) ) ) ) ).

% power2_eq_iff
tff(fact_2363_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_2364_pos2,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% pos2
tff(fact_2365_double__not__eq__Suc__double,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2) != 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)) ).

% double_not_eq_Suc_double
tff(fact_2366_Suc__double__not__eq__double,axiom,
    ! [M2: nat,N: 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))),M2)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% Suc_double_not_eq_double
tff(fact_2367_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_2368_less__exp,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% less_exp
tff(fact_2369_pochhammer__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & comm_semiring_1(A) )
     => ! [X: real,N: nat] : comm_s3205402744901411588hammer(A,real_Vector_of_real(A,X),N) = real_Vector_of_real(A,comm_s3205402744901411588hammer(real,X,N)) ) ).

% pochhammer_of_real
tff(fact_2370_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_2371_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_2372_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_2373_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
             => ( X = Y2 ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2374_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) ) ) ) ).

% power2_le_imp_le
tff(fact_2375_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),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_2376_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_2377_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_2378_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_2379_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = X ) ).

% field_sum_of_halves
tff(fact_2380_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_2381_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_2382_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_2383_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2384_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,bool)),X: A] :
          ( ! [X2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,X2),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(A,A,abs_abs(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_sqrt_wlog
tff(fact_2385_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_2386_nat__induct2,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct2
tff(fact_2387_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),N))) ).

% two_realpow_ge_one
tff(fact_2388_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,N)),semiring_char_0_fact(real,N))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ).

% square_fact_le_2_fact
tff(fact_2389_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),U),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% realpow_square_minus_le
tff(fact_2390_diff__le__diff__pow,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N)))) ) ).

% diff_le_diff_pow
tff(fact_2391_ln__2__less__1,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),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_2392_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),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))),N)),zero_zero(int))) ).

% not_exp_less_eq_0_int
tff(fact_2393_log2__of__power__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) )
     => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M2)) ) ) ).

% log2_of_power_eq
tff(fact_2394_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2)) ) ) ) ).

% power2_less_imp_less
tff(fact_2395_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_2396_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% half_gt_zero_iff
tff(fact_2397_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% sum_power2_ge_zero
tff(fact_2398_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y2 = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2399_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A))) ) ).

% not_sum_power2_lt_zero
tff(fact_2400_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
        <=> ( ( X != zero_zero(A) )
            | ( Y2 != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2401_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ).

% field_less_half_sum
tff(fact_2402_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))) ) ) ) ).

% square_le_1
tff(fact_2403_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y2)) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2404_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))) ) ).

% of_nat_less_two_power
tff(fact_2405_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: 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),M2),N)) != 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))),M2) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2406_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: 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),M2),N)) != 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))),N) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2407_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),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))),N)))) ) ).

% zero_le_even_power'
tff(fact_2408_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,M2: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != 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),N),M2)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2409_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_le_1
tff(fact_2410_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_less_1
tff(fact_2411_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M2: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = 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),M2),N))) ) ).

% div_exp_eq
tff(fact_2412_ex__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N))
          & pp(aa(nat,bool,P,M5)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X4)) ) ) ).

% ex_nat_less
tff(fact_2413_all__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N))
         => pp(aa(nat,bool,P,M5)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X4)) ) ) ).

% all_nat_less
tff(fact_2414_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)) = 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))),N)) ) ).

% minus_power_mult_self
tff(fact_2415_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: 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))),N))) = 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),N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% power_odd_eq
tff(fact_2416_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Z)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% exp_double
tff(fact_2417_nat__bit__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => pp(aa(nat,bool,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_2418_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,X) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_2419_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2420_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% div_2_gt_zero
tff(fact_2421_of__real__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: real] : real_Vector_of_real(A,aa(real,real,exp(real),X)) = aa(A,A,exp(A),real_Vector_of_real(A,X)) ) ).

% of_real_exp
tff(fact_2422_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_Bit0
tff(fact_2423_exp__half__le2,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% exp_half_le2
tff(fact_2424_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) ) ).

% power_minus_Bit0
tff(fact_2425_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_2426_exp__plus__inverse__exp,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X))))) ).

% exp_plus_inverse_exp
tff(fact_2427_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_2428_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_2429_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_2430_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_2431_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_2432_Suc__nat__number__of__add,axiom,
    ! [V2: num,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),one2))),N) ).

% Suc_nat_number_of_add
tff(fact_2433_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_2434_triangle__def,axiom,
    ! [N: nat] : nat_triangle(N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% triangle_def
tff(fact_2435_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2436_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)),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),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),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))),X)),Y2)) ) ).

% power2_diff
tff(fact_2437_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_2438_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))),N)))))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2439_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),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))),N)))),zero_zero(A))) ) ) ).

% odd_power_less_zero
tff(fact_2440_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_2441_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat,M2: 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))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)) = 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),N),M2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% div_exp_mod_exp_eq
tff(fact_2442_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
       => ? [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl2
tff(fact_2443_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
       => ? [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl1
tff(fact_2444_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),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),X),aa(real,real,inverse_inverse(real),X)))) ) ).

% plus_inverse_ge_2
tff(fact_2445_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ).

% exp_bound_half
tff(fact_2446_int__bit__induct,axiom,
    ! [P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,P,zero_zero(int)))
     => ( pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),one_one(int))))
       => ( ! [K2: int] :
              ( pp(aa(int,bool,P,K2))
             => ( ( K2 != zero_zero(int) )
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) ) )
         => ( ! [K2: int] :
                ( pp(aa(int,bool,P,K2))
               => ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) )
           => pp(aa(int,bool,P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_2447_less__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N)),M2))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M2)))) ) ).

% less_log2_of_power
tff(fact_2448_le__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N)),M2))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M2)))) ) ).

% le_log2_of_power
tff(fact_2449_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arsinh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),powr(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),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_2450_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y2: real,X: real] :
          ( ( Y2 != zero_zero(real) )
         => ( real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,X)),real_Vector_of_real(A,Y2)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_2451_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2452_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arcosh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),powr(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),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_2453_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( 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))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = 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),N),M2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2454_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,cosh(A),X) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_2455_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q5: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q5))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q5)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2456_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : aa(A,A,cosh(A),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% cosh_field_def
tff(fact_2457_num_Osize_I5_J,axiom,
    ! [X23: num] : aa(num,nat,size_size(num),aa(num,num,bit0,X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_2458_log2__of__power__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_less
tff(fact_2459_exp__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,exp(real),X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% exp_bound
tff(fact_2460_pos__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_2461_neg__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_2462_pos__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_2463_real__le__x__sinh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ).

% real_le_x_sinh
tff(fact_2464_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_2465_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_2466_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_2467_real__le__abs__sinh,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% real_le_abs_sinh
tff(fact_2468_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_2469_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,X: A,Y2: 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),X),Y2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ) ) ).

% arith_geo_mean
tff(fact_2470_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)) = modulo_modulo(A,X,M2) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M2)),M2) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2471_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2472_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,real_V7770717601297561774m_norm(A,X))),one_one(A))))) ) ).

% norm_less_p1
tff(fact_2473_log2__of__power__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_le
tff(fact_2474_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),real_V7770717601297561774m_norm(A,Z))))) ) ) ).

% exp_bound_lemma
tff(fact_2475_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,exp(real),X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_2476_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,exp(real),X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_2477_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),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)),X)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_2478_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A] : aa(A,A,artanh(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),X)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% artanh_def
tff(fact_2479_neg__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_2480_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% numeral_inc
tff(fact_2481_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,cosh(real),aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% cosh_ln_real
tff(fact_2482_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( archim6421214686448440834_floor(real,aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log2(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),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_2483_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),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))),N))),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))),N))),semiring_char_0_fact(A,N)) ) ).

% fact_double
tff(fact_2484_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),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)),X))),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2485_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,arctan,X)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X)),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),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% arctan_double
tff(fact_2486_tanh__real__altdef,axiom,
    ! [X: real] : aa(real,real,tanh(real),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X)))) ).

% tanh_real_altdef
tff(fact_2487_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2488_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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))),N)))),comm_s3205402744901411588hammer(A,Z,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N)) ) ).

% pochhammer_double
tff(fact_2489_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [B2: real,A3: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),real_Vector_of_real(A,B2)),real_Vector_of_real(A,A3)))),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)))) ) ).

% norm_of_real_diff
tff(fact_2490_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,uminus_uminus(real),X)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),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)),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2491_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => pp(aa(real,bool,aa(real,fun(real,bool),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)),X))),X))),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),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2492_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),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),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2493_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
         => ( archim6421214686448440834_floor(real,aa(real,real,log2(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_2494_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log2(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) )
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_2495_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( archimedean_ceiling(real,aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log2(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),N),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_2496_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),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)),X))),X))),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),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2497_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
         => ( archimedean_ceiling(real,aa(real,real,log2(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_2498_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: 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_2499_low__inv,axiom,
    ! [X: nat,N: nat,Y2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N)))
     => ( vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),X),N) = X ) ) ).

% low_inv
tff(fact_2500_high__inv,axiom,
    ! [X: nat,N: nat,Y2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N)))
     => ( vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),X),N) = Y2 ) ) ).

% high_inv
tff(fact_2501_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,M2: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X2,N) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N)) ) ) ).

% set_n_deg_not_0
tff(fact_2502_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Ma,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2))) ) ).

% high_bound_aux
tff(fact_2503_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: 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_2504_bit__split__inv,axiom,
    ! [X: nat,D2: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(X,D2),vEBT_VEBT_low(X,D2),D2) = X ).

% bit_split_inv
tff(fact_2505_high__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_high(X,N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% high_def
tff(fact_2506_low__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_low(X,N) = modulo_modulo(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% low_def
tff(fact_2507_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se2638667681897837118et_bit(int,N,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2508_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5668285175392031749et_bit(int,N,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2509_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2638667681897837118et_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% unset_bit_negative_int_iff
tff(fact_2510_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5668285175392031749et_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% set_bit_negative_int_iff
tff(fact_2511_divide__complex__def,axiom,
    ! [X: complex,Y2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y2) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),aa(complex,complex,inverse_inverse(complex),Y2)) ).

% divide_complex_def
tff(fact_2512_unset__bit__nat__def,axiom,
    ! [M2: nat,N: nat] : bit_se2638667681897837118et_bit(nat,M2,N) = aa(int,nat,nat2,bit_se2638667681897837118et_bit(int,M2,aa(nat,int,semiring_1_of_nat(int),N))) ).

% unset_bit_nat_def
tff(fact_2513_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_pos_if_in_set
tff(fact_2514_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_2515_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_low(X,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2516_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X2,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_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_1)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_2517_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X2,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_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_1)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_2518_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,N),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_se2638667681897837118et_bit(A,N,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_2519_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,N),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_se5668285175392031749et_bit(A,N,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_2520_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),Deg)))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
         => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_2521_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N),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,N,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_2522_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) )
          & ( ( N != zero_zero(nat) )
           => ( bit_ri4674362597316999326ke_bit(A,N,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_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),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_2523_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),Y2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))
           => ( archimedean_round(A,X) = Y2 ) ) ) ) ).

% round_unique
tff(fact_2524_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_2525_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archimedean_ceiling(A,X) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).

% round_altdef
tff(fact_2526_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),N: nat] :
      ( ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X2)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) ) ) ).

% inthall
tff(fact_2527_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,N,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2528_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se8732182000553998342ip_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% flip_bit_negative_int_iff
tff(fact_2529_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_ri4674362597316999326ke_bit(A,N,zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2530_round__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: int] : archimedean_round(A,aa(int,A,ring_1_of_int(A),N)) = N ) ).

% round_of_int
tff(fact_2531_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_2532_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_ri4674362597316999326ke_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% signed_take_bit_of_minus_1
tff(fact_2533_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_2534_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : 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_2535_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2536_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),N)) = aa(num,int,numeral_numeral(int),N) ) ).

% round_numeral
tff(fact_2537_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_2538_round__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: nat] : archimedean_round(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% round_of_nat
tff(fact_2539_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_2540_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_2541_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N,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_2542_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)) ) ).

% round_neg_numeral
tff(fact_2543_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_2544_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),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),bit_ri4674362597316999326ke_bit(int,N,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_2545_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: 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_2546_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : bit_ri4674362597316999326ke_bit(int,N,aa(int,int,uminus_uminus(int),bit_ri4674362597316999326ke_bit(int,N,K))) = bit_ri4674362597316999326ke_bit(int,N,aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_2547_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : bit_ri4674362597316999326ke_bit(int,N,aa(int,int,aa(int,fun(int,int),minus_minus(int),bit_ri4674362597316999326ke_bit(int,N,K)),bit_ri4674362597316999326ke_bit(int,N,L))) = bit_ri4674362597316999326ke_bit(int,N,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% signed_take_bit_diff
tff(fact_2548_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys2),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2549_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,bool))] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K))
         => ? [X_13: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,I2),X_13)) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K))
             => pp(aa(A,bool,aa(nat,fun(A,bool),P,I2),aa(nat,A,nth(A,Xs3),I2))) ) ) ) ).

% Skolem_list_nth
tff(fact_2550_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2551_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% dbl_def
tff(fact_2552_nth__mem,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,nth(A,Xs),N)),aa(list(A),set(A),set2(A),Xs))) ) ).

% nth_mem
tff(fact_2553_list__ball__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X2)) )
       => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2554_in__set__conv__nth,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( aa(nat,A,nth(A,Xs),I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_2555_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),X: A] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => pp(aa(A,bool,P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_2556_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) )
    <=> ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I2))) ) ) ).

% all_set_conv_all_nth
tff(fact_2557_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y2))) ) ) ).

% round_mono
tff(fact_2558_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_round(A,X))) ) ).

% floor_le_round
tff(fact_2559_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_ceiling(A,X))) ) ).

% ceiling_ge_round
tff(fact_2560_nth__rotate1,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,rotate1(A,Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_2561_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_ri4674362597316999326ke_bit(int,N,K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_2562_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M2: int] : pp(aa(A,bool,aa(A,fun(A,bool),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),M2))))) ) ).

% round_diff_minimal
tff(fact_2563_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_ri4674362597316999326ke_bit(int,N,K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2564_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),bit_ri4674362597316999326ke_bit(int,N,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2565_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),bit_ri4674362597316999326ke_bit(int,N,K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2566_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_ri4674362597316999326ke_bit(int,N,K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2567_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),bit_ri4674362597316999326ke_bit(int,N,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2568_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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))),N)),K))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_ri4674362597316999326ke_bit(int,N,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,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2569_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( bit_ri4674362597316999326ke_bit(int,N,K) = K )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),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))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2570_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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))),N)))
       => ( bit_ri4674362597316999326ke_bit(int,N,K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2571_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N)))),bit_ri4674362597316999326ke_bit(int,N,K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2572_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% round_def
tff(fact_2573_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N),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_ri4674362597316999326ke_bit(A,N,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_2574_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% of_int_round_le
tff(fact_2575_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_ge
tff(fact_2576_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_gt
tff(fact_2577_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),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,X))),X))),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_2578_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),N)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
         => ( archimedean_round(A,X) = N ) ) ) ).

% round_unique'
tff(fact_2579_in__children__def,axiom,
    ! [N: nat,TreeList: list(vEBT_VEBT),X: nat] :
      ( vEBT_V5917875025757280293ildren(N,TreeList,X)
    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,N))),vEBT_VEBT_low(X,N))) ) ).

% in_children_def
tff(fact_2580_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq1
tff(fact_2581_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))),N)),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),N)))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_2582_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N,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_2583_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : bit_concat_bit(aa(nat,nat,suc,N),K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),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))),bit_concat_bit(N,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_2584_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),X)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(real,real,arctan,X)) ) ) ).

% arctan_inverse
tff(fact_2585_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_2586_binomial__Suc__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),N) = aa(nat,nat,suc,N) ).

% binomial_Suc_n
tff(fact_2587_binomial__n__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).

% binomial_n_n
tff(fact_2588_concat__bit__0,axiom,
    ! [K: int,L: int] : bit_concat_bit(zero_zero(nat),K,L) = L ).

% concat_bit_0
tff(fact_2589_binomial__1,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(nat,nat,suc,zero_zero(nat))) = N ).

% binomial_1
tff(fact_2590_binomial__0__Suc,axiom,
    ! [K: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2591_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa(nat,nat,binomial(N),K) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ).

% binomial_eq_0_iff
tff(fact_2592_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_2593_binomial__n__0,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2594_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_concat_bit(N,K,L)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_2595_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_concat_bit(N,K,L)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ).

% concat_bit_negative_iff
tff(fact_2596_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_2597_zdiv__numeral__Bit1,axiom,
    ! [V2: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,V2))),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),V2)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_2598_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).

% zero_less_binomial_iff
tff(fact_2599_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_2600_div__Suc__eq__div__add3,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% div_Suc_eq_div_add3
tff(fact_2601_Suc__div__eq__add3__div__numeral,axiom,
    ! [M2: nat,V2: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M2)))),aa(num,nat,numeral_numeral(nat),V2)) = 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))),M2)),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2602_mod__Suc__eq__mod__add3,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,M2,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = modulo_modulo(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_2603_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M2: nat,V2: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M2))),aa(num,nat,numeral_numeral(nat),V2)) = 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))),M2),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2604_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_2605_zmod__numeral__Bit1,axiom,
    ! [V2: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V2)),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),V2),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_2606_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N,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_2607_choose__one,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).

% choose_one
tff(fact_2608_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X32: num] : aa(num,num,bit0,X23) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_2609_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_2610_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

% pi_neq_zero
tff(fact_2611_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
     => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2612_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K)) ).

% Suc_times_binomial
tff(fact_2613_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).

% Suc_times_binomial_eq
tff(fact_2614_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ).

% binomial_symmetric
tff(fact_2615_binomial__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) ) ).

% binomial_gbinomial
tff(fact_2616_num_Oexhaust,axiom,
    ! [Y2: num] :
      ( ( Y2 != one2 )
     => ( ! [X24: num] : Y2 != aa(num,num,bit0,X24)
       => ~ ! [X33: num] : Y2 != aa(num,num,bit1,X33) ) ) ).

% num.exhaust
tff(fact_2617_pi__not__less__zero,axiom,
    ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),zero_zero(real))) ).

% pi_not_less_zero
tff(fact_2618_pi__gt__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),pi)) ).

% pi_gt_zero
tff(fact_2619_pi__ge__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),pi)) ).

% pi_ge_zero
tff(fact_2620_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(aa(num,num,bit0,X)) = aa(num,num,bit1,X) ).

% inc.simps(2)
tff(fact_2621_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa(num,num,bit1,X)) = aa(num,num,bit0,inc(X)) ).

% inc.simps(3)
tff(fact_2622_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K))) ) ).

% zero_less_binomial
tff(fact_2623_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_2624_choose__mult,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),M2)),aa(nat,nat,binomial(M2),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K))) ) ) ) ).

% choose_mult
tff(fact_2625_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K))),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_2626_binomial__absorb__comp,axiom,
    ! [N: 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),N),K)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_2627_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_2628_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_2629_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_2630_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_2631_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),N) ) ).

% numeral_Bit1_div_2
tff(fact_2632_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),aa(nat,nat,binomial(N),K)) = semiring_char_0_fact(nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_2633_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q5: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q5))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2634_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_2635_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_2636_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N) ).

% Suc3_eq_add_3
tff(fact_2637_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_2638_pi__less__4,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) ).

% pi_less_4
tff(fact_2639_pi__ge__two,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) ).

% pi_ge_two
tff(fact_2640_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_2641_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2642_binomial__mono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),K7))) ) ) ).

% binomial_mono
tff(fact_2643_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),N))) ).

% binomial_maximum'
tff(fact_2644_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% binomial_maximum
tff(fact_2645_binomial__antimono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),K))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K7),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K7)),aa(nat,nat,binomial(N),K))) ) ) ) ).

% binomial_antimono
tff(fact_2646_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),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),N),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2647_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2648_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_2649_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_2650_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,Q5: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q5))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q5))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),Q5)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2651_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q5: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q5))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q5))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q5)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2652_Suc__div__eq__add3__div,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M2)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M2)),N) ).

% Suc_div_eq_add3_div
tff(fact_2653_Suc__mod__eq__add3__mod,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M2))),N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M2),N) ).

% Suc_mod_eq_add3_mod
tff(fact_2654_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_2655_pi__half__less__two,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),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_2656_pi__half__le__two,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),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_2657_exp__le,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,exp(real),one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) ).

% exp_le
tff(fact_2658_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),K7))) ) ) ).

% binomial_strict_mono
tff(fact_2659_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K7),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,binomial(N),K7)),aa(nat,nat,binomial(N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_2660_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_2661_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_2662_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ) ).

% binomial_fact
tff(fact_2663_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).

% fact_binomial
tff(fact_2664_mod__exhaust__less__4,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,M2,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,M2,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_2665_pi__half__gt__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% pi_half_gt_zero
tff(fact_2666_pi__half__ge__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% pi_half_ge_zero
tff(fact_2667_m2pi__less__pi,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_2668_arctan__ubound,axiom,
    ! [Y2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,Y2)),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_2669_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_2670_choose__two,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),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),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% choose_two
tff(fact_2671_minus__pi__half__less__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),zero_zero(real))) ).

% minus_pi_half_less_zero
tff(fact_2672_arctan__lbound,axiom,
    ! [Y2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y2))) ).

% arctan_lbound
tff(fact_2673_arctan__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y2)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,Y2)),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_2674_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq2
tff(fact_2675_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
       => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),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(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),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(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),one_one(nat)),N,one_one(nat))),semiring_char_0_fact(nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_2676_sin__cos__npi,axiom,
    ! [N: nat] : aa(real,real,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))),N)))),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))),N) ).

% sin_cos_npi
tff(fact_2677_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : 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),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_2678_cos__pi__eq__zero,axiom,
    ! [M2: nat] : aa(real,real,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))),M2))))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_2679_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : 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),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_2680_cot__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cot(real),X)),zero_zero(real))) ) ) ).

% cot_less_zero
tff(fact_2681_sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,sin(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sin_zero
tff(fact_2682_cos__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,cos(A),aa(A,A,uminus_uminus(A),X)) = aa(A,A,cos(A),X) ) ).

% cos_minus
tff(fact_2683_sin__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,sin(A),aa(A,A,uminus_uminus(A),X)) = aa(A,A,uminus_uminus(A),aa(A,A,sin(A),X)) ) ).

% sin_minus
tff(fact_2684_cot__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,cot(A),zero_zero(A)) = zero_zero(A) ) ) ).

% cot_zero
tff(fact_2685_cot__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,cot(A),aa(A,A,uminus_uminus(A),X)) = aa(A,A,uminus_uminus(A),aa(A,A,cot(A),X)) ) ).

% cot_minus
tff(fact_2686_cos__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,cos(A),zero_zero(A)) = one_one(A) ) ) ).

% cos_zero
tff(fact_2687_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2688_sin__pi,axiom,
    aa(real,real,sin(real),pi) = zero_zero(real) ).

% sin_pi
tff(fact_2689_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa(nat,nat,suc,N) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( N = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_2690_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,N) )
    <=> ( pred_numeral(K) = N ) ) ).

% eq_numeral_Suc
tff(fact_2691_sin__pi__minus,axiom,
    ! [X: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),X)) = aa(real,real,sin(real),X) ).

% sin_pi_minus
tff(fact_2692_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_2693_cot__pi,axiom,
    aa(real,real,cot(real),pi) = zero_zero(real) ).

% cot_pi
tff(fact_2694_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,sin(A),real_Vector_of_real(A,pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_2695_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_2696_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),pred_numeral(K)),N)) ) ).

% less_numeral_Suc
tff(fact_2697_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),pred_numeral(K))) ) ).

% less_Suc_numeral
tff(fact_2698_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),pred_numeral(K))) ) ).

% le_Suc_numeral
tff(fact_2699_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),pred_numeral(K)),N)) ) ).

% le_numeral_Suc
tff(fact_2700_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_2701_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K)),N) ).

% diff_numeral_Suc
tff(fact_2702_cos__pi,axiom,
    aa(real,real,cos(real),pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_2703_cos__periodic__pi2,axiom,
    ! [X: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),X)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),X)) ).

% cos_periodic_pi2
tff(fact_2704_cos__periodic__pi,axiom,
    ! [X: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),X)) ).

% cos_periodic_pi
tff(fact_2705_sin__periodic__pi2,axiom,
    ! [X: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),X)) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),X)) ).

% sin_periodic_pi2
tff(fact_2706_sin__periodic__pi,axiom,
    ! [X: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),X)) ).

% sin_periodic_pi
tff(fact_2707_cos__minus__pi,axiom,
    ! [X: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),X)) ).

% cos_minus_pi
tff(fact_2708_cos__pi__minus,axiom,
    ! [X: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),X)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),X)) ).

% cos_pi_minus
tff(fact_2709_sin__minus__pi,axiom,
    ! [X: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),X)) ).

% sin_minus_pi
tff(fact_2710_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),X)),aa(A,A,cos(A),X))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),X)),aa(A,A,sin(A),X))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_2711_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,cos(A),real_Vector_of_real(A,pi)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% cos_of_real_pi
tff(fact_2712_sin__npi2,axiom,
    ! [N: nat] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = zero_zero(real) ).

% sin_npi2
tff(fact_2713_sin__npi,axiom,
    ! [N: nat] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% sin_npi
tff(fact_2714_sin__npi__int,axiom,
    ! [N: int] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = zero_zero(real) ).

% sin_npi_int
tff(fact_2715_cot__npi,axiom,
    ! [N: nat] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_2716_cos__pi__half,axiom,
    aa(real,real,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_2717_sin__two__pi,axiom,
    aa(real,real,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_2718_sin__pi__half,axiom,
    aa(real,real,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_2719_cos__two__pi,axiom,
    aa(real,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))),pi)) = one_one(real) ).

% cos_two_pi
tff(fact_2720_cos__periodic,axiom,
    ! [X: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = aa(real,real,cos(real),X) ).

% cos_periodic
tff(fact_2721_sin__periodic,axiom,
    ! [X: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = aa(real,real,sin(real),X) ).

% sin_periodic
tff(fact_2722_cos__2pi__minus,axiom,
    ! [X: real] : aa(real,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)),X)) = aa(real,real,cos(real),X) ).

% cos_2pi_minus
tff(fact_2723_cos__npi,axiom,
    ! [N: nat] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi
tff(fact_2724_cos__npi2,axiom,
    ! [N: nat] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi2
tff(fact_2725_cot__periodic,axiom,
    ! [X: real] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = aa(real,real,cot(real),X) ).

% cot_periodic
tff(fact_2726_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),X)),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,cos(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_2727_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),X)),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,sin(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_2728_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,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_2729_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,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_2730_sin__2npi,axiom,
    ! [N: nat] : aa(real,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))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_2731_cos__2npi,axiom,
    ! [N: nat] : aa(real,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))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_2732_sin__2pi__minus,axiom,
    ! [X: real] : aa(real,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)),X)) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),X)) ).

% sin_2pi_minus
tff(fact_2733_sin__int__2pin,axiom,
    ! [N: int] : aa(real,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)),aa(int,real,ring_1_of_int(real),N))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_2734_cos__int__2pin,axiom,
    ! [N: int] : aa(real,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)),aa(int,real,ring_1_of_int(real),N))) = one_one(real) ).

% cos_int_2pin
tff(fact_2735_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : 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),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_2736_cos__3over2__pi,axiom,
    aa(real,real,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_2737_sin__3over2__pi,axiom,
    aa(real,real,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_2738_cos__of__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: real] : aa(A,A,cos(A),real_Vector_of_real(A,X)) = real_Vector_of_real(A,aa(real,real,cos(real),X)) ) ).

% cos_of_real
tff(fact_2739_cot__of__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: real] : real_Vector_of_real(A,aa(real,real,cot(real),X)) = aa(A,A,cot(A),real_Vector_of_real(A,X)) ) ).

% cot_of_real
tff(fact_2740_sin__of__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: real] : aa(A,A,sin(A),real_Vector_of_real(A,X)) = real_Vector_of_real(A,aa(real,real,sin(real),X)) ) ).

% sin_of_real
tff(fact_2741_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: A] : aa(A,A,cot(A),X3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,cos(A),X3)),aa(A,A,sin(A),X3)) ) ).

% cot_def
tff(fact_2742_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,cos(A),X) = one_one(A) )
         => ( aa(A,A,sin(A),X) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_2743_polar__Ex,axiom,
    ! [X: real,Y2: real] :
    ? [R3: real,A6: real] :
      ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),R3),aa(real,real,cos(real),A6)) )
      & ( Y2 = aa(real,real,aa(real,fun(real,real),times_times(real),R3),aa(real,real,sin(real),A6)) ) ) ).

% polar_Ex
tff(fact_2744_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = 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,sin(A),X)),aa(A,A,cos(A),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),X)),aa(A,A,sin(A),Y2))) ) ).

% sin_add
tff(fact_2745_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),X)),aa(A,A,cos(A),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),X)),aa(A,A,sin(A),Y2))) ) ).

% sin_diff
tff(fact_2746_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) = 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,cos(A),X)),aa(A,A,cos(A),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),X)),aa(A,A,sin(A),Y2))) ) ).

% cos_diff
tff(fact_2747_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),X)),aa(A,A,cos(A),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),X)),aa(A,A,sin(A),Y2))) ) ).

% cos_add
tff(fact_2748_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,sin(A),X) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,cos(A),X)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_2749_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( aa(real,real,sin(real),X) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),aa(real,real,cos(real),X)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_2750_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,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))),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,sin(A),X))),aa(A,A,cos(A),X)) ) ).

% sin_double
tff(fact_2751_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y3))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),pi))
      & ( aa(real,real,sin(real),Y3) = aa(real,real,sin(real),X) )
      & ( aa(real,real,cos(real),Y3) = aa(real,real,cos(real),X) ) ) ).

% sincos_principal_value
tff(fact_2752_sin__x__le__x,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sin(real),X)),X)) ) ).

% sin_x_le_x
tff(fact_2753_sin__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sin(real),X)),one_one(real))) ).

% sin_le_one
tff(fact_2754_cos__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,cos(real),X)),one_one(real))) ).

% cos_le_one
tff(fact_2755_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,sin(real),X))),aa(real,real,abs_abs(real),X))) ).

% abs_sin_x_le_abs_x
tff(fact_2756_cos__arctan__not__zero,axiom,
    ! [X: real] : aa(real,real,cos(real),aa(real,real,arctan,X)) != zero_zero(real) ).

% cos_arctan_not_zero
tff(fact_2757_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_2758_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M2: int,X: real] : aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M2)),real_Vector_of_real(A,X))) = real_Vector_of_real(A,aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M2)),X))) ) ).

% cos_int_times_real
tff(fact_2759_sin__cos__le1,axiom,
    ! [X: real,Y2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sin(real),X)),aa(real,real,sin(real),Y2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,cos(real),X)),aa(real,real,cos(real),Y2))))),one_one(real))) ).

% sin_cos_le1
tff(fact_2760_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M2: int,X: real] : aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M2)),real_Vector_of_real(A,X))) = real_Vector_of_real(A,aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M2)),X))) ) ).

% sin_int_times_real
tff(fact_2761_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),X)),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),aa(A,A,sin(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cos_squared_eq
tff(fact_2762_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),X)),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),aa(A,A,cos(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% sin_squared_eq
tff(fact_2763_sin__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),pi))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sin(real),X))) ) ) ).

% sin_gt_zero
tff(fact_2764_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),X)),aa(real,real,sin(real),X))) ) ).

% sin_x_ge_neg_x
tff(fact_2765_sin__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sin(real),X))) ) ) ).

% sin_ge_zero
tff(fact_2766_sin__ge__minus__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,sin(real),X))) ).

% sin_ge_minus_one
tff(fact_2767_cos__inj__pi,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),pi))
           => ( ( aa(real,real,cos(real),X) = aa(real,real,cos(real),Y2) )
             => ( X = Y2 ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_2768_cos__mono__le__eq,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),pi))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,cos(real),X)),aa(real,real,cos(real),Y2)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),X)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_2769_cos__monotone__0__pi__le,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,cos(real),X)),aa(real,real,cos(real),Y2))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_2770_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),pi)) = zero_zero(real) )
    <=> pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),ring_1_Ints(real))) ) ).

% sin_times_pi_eq_0
tff(fact_2771_cos__ge__minus__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,cos(real),X))) ).

% cos_ge_minus_one
tff(fact_2772_abs__sin__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,sin(real),X))),one_one(real))) ).

% abs_sin_le_one
tff(fact_2773_abs__cos__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,cos(real),X))),one_one(real))) ).

% abs_cos_le_one
tff(fact_2774_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),aa(A,A,cos(A),W)),aa(A,A,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))),aa(A,A,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)))))),aa(A,A,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_2775_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),aa(A,A,sin(A),W)),aa(A,A,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))),aa(A,A,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)))))),aa(A,A,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_2776_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),aa(A,A,sin(A),W)),aa(A,A,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))),aa(A,A,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)))))),aa(A,A,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_2777_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),aa(A,A,cos(A),W)),aa(A,A,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),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),aa(A,A,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_2778_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),aa(A,A,sin(A),W)),aa(A,A,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),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),aa(A,A,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_2779_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),aa(A,A,sin(A),W)),aa(A,A,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),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),aa(A,A,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_2780_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,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))),X)) = 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,cos(A),X)),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,sin(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cos_double
tff(fact_2781_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,sin(A),X) = aa(A,A,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)))),X)) ) ).

% sin_cos_eq
tff(fact_2782_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,cos(A),X) = aa(A,A,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)))),X)) ) ).

% cos_sin_eq
tff(fact_2783_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_2784_cos__double__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : aa(A,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),aa(A,A,sin(A),W)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% cos_double_sin
tff(fact_2785_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,uminus_uminus(A),aa(A,A,sin(A),X)) = aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% minus_sin_cos_eq
tff(fact_2786_cos__two__neq__zero,axiom,
    aa(real,real,cos(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_2787_cos__monotone__0__pi,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cos(real),X)),aa(real,real,cos(real),Y2))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_2788_cos__mono__less__eq,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),pi))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cos(real),X)),aa(real,real,cos(real),Y2)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_2789_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),pi))
       => ( ( aa(real,real,sin(real),X) = zero_zero(real) )
         => ( X = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_2790_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),pi))
     => ( ( aa(real,real,sin(real),X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_2791_cos__monotone__minus__pi__0_H,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,cos(real),Y2)),aa(real,real,cos(real),X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_2792_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( aa(real,real,sin(real),X) = zero_zero(real) )
    <=> ? [I2: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),pi) ) ).

% sin_zero_iff_int2
tff(fact_2793_sincos__total__pi,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
       => ? [T3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T3),pi))
            & ( X = aa(real,real,cos(real),T3) )
            & ( Y2 = aa(real,real,sin(real),T3) ) ) ) ) ).

% sincos_total_pi
tff(fact_2794_sin__expansion__lemma,axiom,
    ! [X: real,M2: nat] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M2))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M2)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% sin_expansion_lemma
tff(fact_2795_cos__expansion__lemma,axiom,
    ! [X: real,M2: nat] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M2))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M2)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% cos_expansion_lemma
tff(fact_2796_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sin(real),X))) ) ) ).

% sin_gt_zero_02
tff(fact_2797_cos__two__less__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cos(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),zero_zero(real))) ).

% cos_two_less_zero
tff(fact_2798_cos__two__le__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,cos(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),zero_zero(real))) ).

% cos_two_le_zero
tff(fact_2799_cos__is__zero,axiom,
    ? [X2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X2))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( aa(real,real,cos(real),X2) = zero_zero(real) )
      & ! [Y: real] :
          ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
            & ( aa(real,real,cos(real),Y) = zero_zero(real) ) )
         => ( Y = X2 ) ) ) ).

% cos_is_zero
tff(fact_2800_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,B2: nat,A3: nat,F2: fun(nat,fun(A,A)),Acc: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
       => ( set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc) = Acc ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
       => ( set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc) = 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),Acc)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2801_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb3: nat,Xc: A,Y2: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb3,Xc) = Y2 )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb3),Xa))
         => ( Y2 = Xc ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb3),Xa))
         => ( Y2 = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb3,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2802_cos__monotone__minus__pi__0,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cos(real),Y2)),aa(real,real,cos(real),X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_2803_cos__total,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => ? [X2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X2))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),pi))
            & ( aa(real,real,cos(real),X2) = Y2 )
            & ! [Y: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),pi))
                  & ( aa(real,real,cos(real),Y) = Y2 ) )
               => ( Y = X2 ) ) ) ) ) ).

% cos_total
tff(fact_2804_sincos__total__pi__half,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
         => ? [T3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T3))
              & pp(aa(real,bool,aa(real,fun(real,bool),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)))))
              & ( X = aa(real,real,cos(real),T3) )
              & ( Y2 = aa(real,real,sin(real),T3) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_2805_sincos__total__2pi__le,axiom,
    ! [X: real,Y2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
     => ? [T3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T3))
          & pp(aa(real,bool,aa(real,fun(real,bool),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)))
          & ( X = aa(real,real,cos(real),T3) )
          & ( Y2 = aa(real,real,sin(real),T3) ) ) ) ).

% sincos_total_2pi_le
tff(fact_2806_sincos__total__2pi,axiom,
    ! [X: real,Y2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
     => ~ ! [T3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),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)))
             => ( ( X = aa(real,real,cos(real),T3) )
               => ( Y2 != aa(real,real,sin(real),T3) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_2807_sin__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( aa(real,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)),N)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_2808_cos__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( aa(real,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)),N)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_2809_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_2810_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),aa(A,A,cos(A),W)),aa(A,A,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))),aa(A,A,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)))))),aa(A,A,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_2811_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),aa(A,A,cos(A),W)),aa(A,A,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),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),aa(A,A,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_2812_sin__gt__zero2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sin(real),X))) ) ) ).

% sin_gt_zero2
tff(fact_2813_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sin(real),X)),zero_zero(real))) ) ) ).

% sin_lt_zero
tff(fact_2814_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,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))),X))),one_one(real))) ) ) ).

% cos_double_less_one
tff(fact_2815_sin__30,axiom,
    aa(real,real,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_2816_cos__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,cos(real),X))) ) ) ).

% cos_gt_zero
tff(fact_2817_sin__monotone__2pi__le,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sin(real),Y2)),aa(real,real,sin(real),X))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_2818_sin__mono__le__eq,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sin(real),X)),aa(real,real,sin(real),Y2)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_2819_sin__inj__pi,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),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,sin(real),X) = aa(real,real,sin(real),Y2) )
             => ( X = Y2 ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_2820_cos__60,axiom,
    aa(real,real,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_2821_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa(real,real,cos(real),X) = one_one(real) )
    <=> ? [X4: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),X4)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_2822_cos__double__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : aa(A,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),aa(A,A,cos(A),W)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_2823_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),X)) = aa(A,A,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),aa(A,A,cos(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),aa(A,A,cos(A),X))) ) ).

% cos_treble_cos
tff(fact_2824_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),pi),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sin(real),X)),zero_zero(real))) ) ) ).

% sin_le_zero
tff(fact_2825_sin__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sin(real),X)),zero_zero(real))) ) ) ).

% sin_less_zero
tff(fact_2826_sin__mono__less__eq,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sin(real),X)),aa(real,real,sin(real),Y2)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_2827_sin__monotone__2pi,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sin(real),Y2)),aa(real,real,sin(real),X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_2828_sin__total,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => ? [X2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X2))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
            & ( aa(real,real,sin(real),X2) = Y2 )
            & ! [Y: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
                  & ( aa(real,real,sin(real),Y) = Y2 ) )
               => ( Y = X2 ) ) ) ) ) ).

% sin_total
tff(fact_2829_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,cos(real),X))) ) ) ).

% cos_gt_zero_pi
tff(fact_2830_cos__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,cos(real),X))) ) ) ).

% cos_ge_zero
tff(fact_2831_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa(real,real,cos(real),X) = one_one(real) )
    <=> ( ? [X4: nat] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X4)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)
        | ? [X4: nat] : X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X4)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_2832_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_2833_cot__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,cot(real),X))) ) ) ).

% cot_gt_zero
tff(fact_2834_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N,one_one(nat))) ) ).

% fact_code
tff(fact_2835_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,cos(A),X) != zero_zero(A) )
         => ( ( aa(A,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))),X)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,tan(A),X))),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),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ) ).

% tan_double
tff(fact_2836_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),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(aa(real,real,cos(real),T3),aa(real,real,sin(real),T3)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_2837_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sin(real),X) = zero_zero(real) )
    <=> ( ? [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5))
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_2838_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,cos(real),X) = zero_zero(real) )
    <=> ( ? [N5: nat] :
            ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N5: nat] :
            ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5))
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_2839_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( aa(real,real,sin(real),X) = zero_zero(real) )
       => ? [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_2840_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( aa(real,real,cos(real),X) = zero_zero(real) )
       => ? [N2: nat] :
            ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_2841_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),zero_zero(A))) ) ).

% dvd_0_right
tff(fact_2842_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A3))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_2843_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_triv_right_iff
tff(fact_2844_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_triv_left_iff
tff(fact_2845_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ) ).

% div_dvd_div
tff(fact_2846_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,uminus_uminus(A),X)),Y2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y2)) ) ) ).

% minus_dvd_iff
tff(fact_2847_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(A,A,uminus_uminus(A),Y2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y2)) ) ) ).

% dvd_minus_iff
tff(fact_2848_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M2),aa(A,A,abs_abs(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M2),K)) ) ) ).

% dvd_abs_iff
tff(fact_2849_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,abs_abs(A),M2)),K))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M2),K)) ) ) ).

% abs_dvd_iff
tff(fact_2850_tan__pi,axiom,
    aa(real,real,tan(real),pi) = zero_zero(real) ).

% tan_pi
tff(fact_2851_tan__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tan(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tan_zero
tff(fact_2852_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),one_one(nat)))
    <=> ( M2 = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_2853_tan__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),aa(A,A,uminus_uminus(A),X)) = aa(A,A,uminus_uminus(A),aa(A,A,tan(A),X)) ) ).

% tan_minus
tff(fact_2854_tan__periodic__pi,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),pi)) = aa(real,real,tan(real),X) ).

% tan_periodic_pi
tff(fact_2855_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2856_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2857_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_2858_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_2859_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))) ) ) ) ).

% unit_prod
tff(fact_2860_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2861_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2862_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2863_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2864_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2865_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_2866_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A))) ) ) ) ).

% unit_div
tff(fact_2867_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_2868_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),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_2869_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( modulo_modulo(A,B2,A3) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_2870_dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( M2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_2871_dvd__1__left,axiom,
    ! [K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K)) ).

% dvd_1_left
tff(fact_2872_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( ( K = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_2873_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2874_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2875_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% pow_divides_pow_iff
tff(fact_2876_tan__npi,axiom,
    ! [N: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_2877_tan__periodic__n,axiom,
    ! [X: real,N: num] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),N)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_n
tff(fact_2878_tan__periodic__nat,axiom,
    ! [X: real,N: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_nat
tff(fact_2879_tan__periodic__int,axiom,
    ! [X: real,I: int] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_int
tff(fact_2880_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,N))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_2881_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ) ).

% even_Suc
tff(fact_2882_norm__cos__sin,axiom,
    ! [T2: real] : real_V7770717601297561774m_norm(complex,complex2(aa(real,real,cos(real),T2),aa(real,real,sin(real),T2))) = one_one(real) ).

% norm_cos_sin
tff(fact_2883_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) ) ) ).

% even_plus_one_iff
tff(fact_2884_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2885_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_2886_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A3: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).

% power_minus_odd
tff(fact_2887_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_2888_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ).

% even_Suc_div_two
tff(fact_2889_tan__periodic,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic
tff(fact_2890_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2891_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2892_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),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_2893_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W)))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2894_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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),N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% even_power
tff(fact_2895_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2896_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq
tff(fact_2897_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_2898_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_2899_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ) ) ).

% even_of_nat
tff(fact_2900_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_2901_even__diff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),M2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),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),M2),N))) ) ) ).

% even_diff_nat
tff(fact_2902_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),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_2903_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))),N)),one_one(A))))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2904_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2905_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( 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),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_2906_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W)))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_2907_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = 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))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_2908_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = 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))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2909_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_trans
tff(fact_2910_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),A3)) ) ).

% dvd_refl
tff(fact_2911_dvd__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2))
       => ( M2 = N ) ) ) ).

% dvd_antisym
tff(fact_2912_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ) ).

% of_nat_dvd_iff
tff(fact_2913_tan__of__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: real] : real_Vector_of_real(A,aa(real,real,tan(real),X)) = aa(A,A,tan(A),real_Vector_of_real(A,X)) ) ).

% tan_of_real
tff(fact_2914_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A3))
         => ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_2915_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
        <=> ( ( A3 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_2916_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3))) ) ).

% dvd_triv_right
tff(fact_2917_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ).

% dvd_mult_right
tff(fact_2918_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_2919_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ).

% dvd_triv_left
tff(fact_2920_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ).

% dvd_mult_left
tff(fact_2921_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult2
tff(fact_2922_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult
tff(fact_2923_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
        <=> ? [K3: A] : A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvd_def
tff(fact_2924_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) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% dvdI
tff(fact_2925_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ~ ! [K2: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvdE
tff(fact_2926_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),one_one(A)),A3)) ) ).

% one_dvd
tff(fact_2927_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% unit_imp_dvd
tff(fact_2928_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_2929_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_add_right_iff
tff(fact_2930_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_add_left_iff
tff(fact_2931_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ) ).

% dvd_add
tff(fact_2932_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y2),Z))) ) ) ) ).

% dvd_diff
tff(fact_2933_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).

% dvd_diff_commute
tff(fact_2934_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2935_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) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
             => ( A3 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2936_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2937_gcd__nat_Oextremum,axiom,
    ! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),zero_zero(nat))) ).

% gcd_nat.extremum
tff(fact_2938_gcd__nat_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
        & ( zero_zero(nat) != A3 ) ) ).

% gcd_nat.extremum_strict
tff(fact_2939_gcd__nat_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
    <=> ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_2940_gcd__nat_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),zero_zero(nat)))
        & ( A3 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_2941_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
     => ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_2942_complex__minus,axiom,
    ! [A3: real,B2: real] : aa(complex,complex,uminus_uminus(complex),complex2(A3,B2)) = complex2(aa(real,real,uminus_uminus(real),A3),aa(real,real,uminus_uminus(real),B2)) ).

% complex_minus
tff(fact_2943_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),K)) ) ) ).

% dvd_if_abs_eq
tff(fact_2944_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),modulo_modulo(A,A3,B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3)) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2945_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),modulo_modulo(A,A3,B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3)) ) ) ) ).

% dvd_mod_iff
tff(fact_2946_dvd__diff__nat,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))) ) ) ).

% dvd_diff_nat
tff(fact_2947_complex__diff,axiom,
    ! [A3: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),complex2(A3,B2)),complex2(C2,D2)) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),C2),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),D2)) ).

% complex_diff
tff(fact_2948_tan__arctan,axiom,
    ! [Y2: real] : aa(real,real,tan(real),aa(real,real,arctan,Y2)) = Y2 ).

% tan_arctan
tff(fact_2949_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),one_one(A))) ) ).

% not_is_unit_0
tff(fact_2950_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z3: B] :
        ! [X3: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Z3))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S))) ) ) ) ).

% minf(10)
tff(fact_2951_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z3: B] :
        ! [X3: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Z3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S))) ) ) ) ).

% minf(9)
tff(fact_2952_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z3: B] :
        ! [X3: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X3))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S))) ) ) ) ).

% pinf(10)
tff(fact_2953_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z3: B] :
        ! [X3: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S))) ) ) ) ).

% pinf(9)
tff(fact_2954_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2955_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).

% is_unit_mult_iff
tff(fact_2956_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_2957_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_2958_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2959_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2960_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2961_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2962_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),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_2963_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2964_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2965_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2966_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ).

% dvd_mult_imp_div
tff(fact_2967_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2968_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_div_unit_iff
tff(fact_2969_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% div_unit_dvd_iff
tff(fact_2970_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2971_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2972_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_2973_Complex__eq__1,axiom,
    ! [A3: real,B2: real] :
      ( ( complex2(A3,B2) = one_one(complex) )
    <=> ( ( A3 = one_one(real) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_1
tff(fact_2974_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_2975_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2976_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2977_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ) ) ).

% div_power
tff(fact_2978_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% mod_0_imp_dvd
tff(fact_2979_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
        <=> ( modulo_modulo(A,B2,A3) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2980_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_2981_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_2982_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) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))) ) ) ).

% mod_eq_dvd_iff
tff(fact_2983_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2)) ) ) ).

% nat_dvd_not_less
tff(fact_2984_dvd__pos__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2)) ) ) ).

% dvd_pos_nat
tff(fact_2985_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ) ).

% dvd_minus_self
tff(fact_2986_less__eq__dvd__minus,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ).

% less_eq_dvd_minus
tff(fact_2987_dvd__diffD1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_2988_dvd__diffD,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M2)) ) ) ) ).

% dvd_diffD
tff(fact_2989_bezout1__nat,axiom,
    ! [A3: nat,B2: nat] :
    ? [D5: nat,X2: nat,Y3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),A3))
      & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = D5 )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)) = D5 ) ) ) ).

% bezout1_nat
tff(fact_2990_cot__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,cot(A),X) = aa(A,A,inverse_inverse(A),aa(A,A,tan(A),X)) ) ).

% cot_altdef
tff(fact_2991_tan__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),X) = aa(A,A,inverse_inverse(A),aa(A,A,cot(A),X)) ) ).

% tan_altdef
tff(fact_2992_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2993_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,bool),L: A] :
          ( ? [X4: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4)))
        <=> ? [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A))))
              & pp(aa(A,bool,P,X4)) ) ) ) ).

% unity_coeff_ex
tff(fact_2994_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) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2995_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2996_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2997_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_2998_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2999_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D6: A,T2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),D6))
         => ! [X3: A,K4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),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),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D6))),T2))) ) ) ) ).

% inf_period(3)
tff(fact_3000_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D6: A,T2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),D6))
         => ! [X3: A,K4: A] :
              ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T2)))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),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),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D6))),T2))) ) ) ) ).

% inf_period(4)
tff(fact_3001_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_3002_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_3003_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_3004_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),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_3005_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_3006_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_3007_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( modulo_modulo(A,A3,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_3008_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_3009_Complex__eq__neg__1,axiom,
    ! [A3: real,B2: real] :
      ( ( complex2(A3,B2) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) )
    <=> ( ( A3 = aa(real,real,uminus_uminus(real),one_one(real)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_1
tff(fact_3010_Complex__eq__neg__numeral,axiom,
    ! [A3: real,B2: real,W: num] :
      ( ( complex2(A3,B2) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) )
    <=> ( ( A3 = aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_3011_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).

% dvd_imp_le
tff(fact_3012_dvd__mult__cancel,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ) ).

% dvd_mult_cancel
tff(fact_3013_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3014_bezout__add__strong__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [D5: nat,X2: nat,Y3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),A3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),B2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D5) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3015_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M2,N)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3016_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M2: nat,Q5: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( ( modulo_modulo(nat,M2,Q5) = modulo_modulo(nat,N,Q5) )
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Q5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_3017_real__of__nat__div,axiom,
    ! [D2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),N))
     => ( aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),D2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),D2)) ) ) ).

% real_of_nat_div
tff(fact_3018_dvd__fact,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),semiring_char_0_fact(nat,N))) ) ) ).

% dvd_fact
tff(fact_3019_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: A] : aa(A,A,tan(A),X3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sin(A),X3)),aa(A,A,cos(A),X3)) ) ).

% tan_def
tff(fact_3020_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),zero_zero(A))) ) ).

% even_zero
tff(fact_3021_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [B4: A] :
                  ( ( B4 != zero_zero(A) )
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B4),one_one(A)))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = B4 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B4) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B4) = 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),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3022_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_3023_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_3024_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),one_one(A))) ) ).

% odd_one
tff(fact_3025_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),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_3026_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) ) ) ).

% even_minus
tff(fact_3027_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M2: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A)))
              | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ) ).

% dvd_power_iff
tff(fact_3028_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            | ( X = one_one(A) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))) ) ) ).

% dvd_power
tff(fact_3029_div2__even__ext__nat,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Y2)) )
       => ( X = Y2 ) ) ) ).

% div2_even_ext_nat
tff(fact_3030_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),semiring_char_0_fact(A,N))) ) ) ).

% choose_dvd
tff(fact_3031_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),M2))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_3032_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2)),M2))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_3033_dvd__minus__add,axiom,
    ! [Q5: nat,N: nat,R: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q5),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q5),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M2)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Q5)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M2)),Q5)))) ) ) ) ).

% dvd_minus_add
tff(fact_3034_power__dvd__imp__le,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% power_dvd_imp_le
tff(fact_3035_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),R),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R),M2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),R)))
         => ( modulo_modulo(nat,M2,N) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_3036_complex__mult,axiom,
    ! [A3: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A3,B2)),complex2(C2,D2)) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),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),A3),D2)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),C2))) ).

% complex_mult
tff(fact_3037_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_3038_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_3039_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),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_3040_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% uminus_power_if
tff(fact_3041_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% odd_pos
tff(fact_3042_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2638667681897837118et_bit(A,M2,A3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
            | ( M2 = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_3043_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5668285175392031749et_bit(A,M2,A3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
            & ( M2 != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_3044_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se8732182000553998342ip_bit(A,M2,A3)))
        <=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
            <=> ( M2 = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_3045_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_3046_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ~ ! [B4: 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))),B4)),one_one(A)) ) ) ).

% oddE
tff(fact_3047_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),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) ) )
         => ~ ( ~ pp(aa(A,bool,aa(A,fun(A,bool),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_3048_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),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) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),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) ) ) ) ) ).

% mod2_eq_if
tff(fact_3049_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% zero_le_even_power
tff(fact_3050_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% zero_le_odd_power
tff(fact_3051_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq
tff(fact_3052_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% minus_one_power_iff
tff(fact_3053_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% central_binomial_odd
tff(fact_3054_tan__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,tan(real),X))) ) ) ).

% tan_gt_zero
tff(fact_3055_lemma__tan__total,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y2))
     => ? [X2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X2))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(real,real,tan(real),X2))) ) ) ).

% lemma_tan_total
tff(fact_3056_lemma__tan__total1,axiom,
    ! [Y2: real] :
    ? [X2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X2))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
      & ( aa(real,real,tan(real),X2) = Y2 ) ) ).

% lemma_tan_total1
tff(fact_3057_tan__mono__lt__eq,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y2)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3058_tan__monotone_H,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),X))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),Y2)),aa(real,real,tan(real),X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3059_tan__monotone,axiom,
    ! [Y2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),Y2)),aa(real,real,tan(real),X))) ) ) ) ).

% tan_monotone
tff(fact_3060_tan__total,axiom,
    ! [Y2: real] :
    ? [X2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X2))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
      & ( aa(real,real,tan(real),X2) = Y2 )
      & ! [Y: real] :
          ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
            & pp(aa(real,bool,aa(real,fun(real,bool),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,real,tan(real),Y) = Y2 ) )
         => ( Y = X2 ) ) ) ).

% tan_total
tff(fact_3061_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_3062_tan__inverse,axiom,
    ! [Y2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,tan(real),Y2)) = 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)))),Y2)) ).

% tan_inverse
tff(fact_3063_tan__cot,axiom,
    ! [X: 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)))),X)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),X)) ).

% tan_cot
tff(fact_3064_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
        <=> ( ( N = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
              & ( A3 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq
tff(fact_3065_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,cos(A),X) != zero_zero(A) )
         => ( ( aa(A,A,cos(A),Y2) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),X)),aa(A,A,cos(A),Y2))) ) ) ) ) ).

% add_tan_eq
tff(fact_3066_tan__cot_H,axiom,
    ! [X: 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)))),X)) = aa(real,real,cot(real),X) ).

% tan_cot'
tff(fact_3067_tan__total__pos,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
     => ? [X2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X2))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
          & ( aa(real,real,tan(real),X2) = Y2 ) ) ) ).

% tan_total_pos
tff(fact_3068_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),X))) ) ) ).

% tan_pos_pi2_le
tff(fact_3069_tan__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),X)),zero_zero(real))) ) ) ).

% tan_less_zero
tff(fact_3070_tan__mono__le__eq,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y2)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3071_tan__mono__le,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y2))) ) ) ) ).

% tan_mono_le
tff(fact_3072_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),X))),one_one(real))) ) ).

% tan_bound_pi2
tff(fact_3073_arctan__unique,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),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),X) = Y2 )
         => ( aa(real,real,arctan,Y2) = X ) ) ) ) ).

% arctan_unique
tff(fact_3074_arctan__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),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),X)) = X ) ) ) ).

% arctan_tan
tff(fact_3075_arctan,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y2)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,Y2)),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,Y2)) = Y2 ) ) ).

% arctan
tff(fact_3076_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))),M2)),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))),N))))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% even_mask_div_iff'
tff(fact_3077_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3078_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,cos(A),X) != zero_zero(A) )
         => ( ( aa(A,A,cos(A),Y2) != 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),X)),aa(A,A,tan(A),Y2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),X)),aa(A,A,cos(A),Y2))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3079_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,cos(A),X) != zero_zero(A) )
         => ( ( aa(A,A,cos(A),Y2) != zero_zero(A) )
           => ( ( aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) = 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),X)),aa(A,A,tan(A),Y2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y2)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3080_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,cos(A),X) != zero_zero(A) )
         => ( ( aa(A,A,cos(A),Y2) != zero_zero(A) )
           => ( ( aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y2))),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),X)),aa(A,A,tan(A),Y2)))) ) ) ) ) ) ).

% tan_add
tff(fact_3081_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),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),N),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_3082_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_3083_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ? [Z3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))),Z3))
          & pp(aa(real,bool,aa(real,fun(real,bool),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) = X ) ) ) ).

% tan_total_pi4
tff(fact_3084_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))),M2)),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))),N))))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).

% even_mask_div_iff
tff(fact_3085_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,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)) )
     => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),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),N),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_3086_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_3087_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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))),M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),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),N),M2))))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3088_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,cos(A),X) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),aa(A,A,cos(A),X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ) ).

% tan_sec
tff(fact_3089_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,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))),X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,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))),X))),one_one(A))) ) ).

% tan_half
tff(fact_3090_sin__coeff__def,axiom,
    ! [X3: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X3))
       => ( aa(nat,real,sin_coeff,X3) = zero_zero(real) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X3))
       => ( aa(nat,real,sin_coeff,X3) = 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),X3),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),semiring_char_0_fact(real,X3)) ) ) ) ).

% sin_coeff_def
tff(fact_3091_cos__coeff__def,axiom,
    ! [X3: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X3))
       => ( aa(nat,real,cos_coeff,X3) = 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),X3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),semiring_char_0_fact(real,X3)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X3))
       => ( aa(nat,real,cos_coeff,X3) = zero_zero(real) ) ) ) ).

% cos_coeff_def
tff(fact_3092_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( aa(real,real,sin(real),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,tan(real),X)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% sin_tan
tff(fact_3093_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( aa(real,real,cos(real),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% cos_tan
tff(fact_3094_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
       => ( aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = one_one(real) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
       => ( aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = aa(real,real,uminus_uminus(real),one_one(real)) ) ) ) ).

% cos_npi_int
tff(fact_3095_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_3096_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sqrt,X) = one_one(real) )
    <=> ( X = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_3097_real__sqrt__gt__1__iff,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,sqrt,Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),Y2)) ) ).

% real_sqrt_gt_1_iff
tff(fact_3098_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ).

% real_sqrt_lt_1_iff
tff(fact_3099_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ).

% real_sqrt_le_1_iff
tff(fact_3100_real__sqrt__ge__1__iff,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y2)) ) ).

% real_sqrt_ge_1_iff
tff(fact_3101_int__dvd__int__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ).

% int_dvd_int_iff
tff(fact_3102_zdvd1__eq,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),one_one(int)))
    <=> ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_3103_sin__coeff__0,axiom,
    aa(nat,real,sin_coeff,zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3104_cos__coeff__0,axiom,
    aa(nat,real,cos_coeff,zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3105_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),L)),K))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_3106_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R))),K))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_3107_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),K)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        | ( R = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_3108_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        | ( R = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_3109_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_abs_dvd_iff
tff(fact_3110_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ) ).

% dvd_nat_abs_iff
tff(fact_3111_real__sqrt__divide,axiom,
    ! [X: real,Y2: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y2)) ).

% real_sqrt_divide
tff(fact_3112_real__sqrt__minus,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,X)) ).

% real_sqrt_minus
tff(fact_3113_uminus__dvd__conv_I1_J,axiom,
    ! [D2: int,T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),T2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,uminus_uminus(int),D2)),T2)) ) ).

% uminus_dvd_conv(1)
tff(fact_3114_uminus__dvd__conv_I2_J,axiom,
    ! [D2: int,T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),T2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,uminus_uminus(int),T2))) ) ).

% uminus_dvd_conv(2)
tff(fact_3115_zdvd__zdiffD,axiom,
    ! [K: int,M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),minus_minus(int),M2),N)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),M2)) ) ) ).

% zdvd_zdiffD
tff(fact_3116_zdvd__antisym__abs,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),A3))
       => ( aa(int,int,abs_abs(int),A3) = aa(int,int,abs_abs(int),B2) ) ) ) ).

% zdvd_antisym_abs
tff(fact_3117_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_ge_one
tff(fact_3118_zdvd__antisym__nonneg,axiom,
    ! [M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M2),N))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M2))
           => ( M2 = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3119_zdvd__not__zless,axiom,
    ! [M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M2),N))
       => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M2)) ) ) ).

% zdvd_not_zless
tff(fact_3120_zdvd__mult__cancel,axiom,
    ! [K: int,M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N)))
     => ( ( K != zero_zero(int) )
       => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M2),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_3121_zdvd__mono,axiom,
    ! [K: int,M2: int,T2: int] :
      ( ( K != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M2),T2))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),K),T2))) ) ) ).

% zdvd_mono
tff(fact_3122_zdvd__period,axiom,
    ! [A3: int,D2: int,X: int,T2: int,C2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),D2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D2))),T2))) ) ) ).

% zdvd_period
tff(fact_3123_zdvd__reduce,axiom,
    ! [K: int,N: int,M2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),aa(int,int,aa(int,fun(int,int),times_times(int),K),M2))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),N)) ) ).

% zdvd_reduce
tff(fact_3124_abs__div,axiom,
    ! [Y2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Y2),X))
     => ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),X)),aa(int,int,abs_abs(int),Y2)) ) ) ).

% abs_div
tff(fact_3125_sin__coeff__Suc,axiom,
    ! [N: nat] : aa(nat,real,sin_coeff,aa(nat,nat,suc,N)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,cos_coeff,N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% sin_coeff_Suc
tff(fact_3126_real__div__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,X)) = aa(real,real,sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_3127_cos__coeff__Suc,axiom,
    ! [N: nat] : aa(nat,real,cos_coeff,aa(nat,nat,suc,N)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(nat,real,sin_coeff,N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% cos_coeff_Suc
tff(fact_3128_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z),N))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_3129_dvd__imp__le__int,axiom,
    ! [I: int,D2: int] :
      ( ( I != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),I))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),D2)),aa(int,int,abs_abs(int),I))) ) ) ).

% dvd_imp_le_int
tff(fact_3130_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B2))),ring_1_Ints(A))) ) ) ).

% of_int_divide_in_Ints
tff(fact_3131_real__of__int__div,axiom,
    ! [D2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),N))
     => ( aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),D2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),D2)) ) ) ).

% real_of_int_div
tff(fact_3132_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
       => ( aa(int,int,sgn_sgn(int),modulo_modulo(int,K,L)) = aa(int,int,sgn_sgn(int),L) ) ) ) ).

% sgn_mod
tff(fact_3133_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,X)),X) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_3134_zdvd__mult__cancel1,axiom,
    ! [M2: int,N: int] :
      ( ( M2 != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),M2),N)),M2))
      <=> ( aa(int,int,abs_abs(int),N) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_3135_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        | ( ( L = zero_zero(int) )
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) )
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L)) ) ) ).

% mod_int_pos_iff
tff(fact_3136_aset_I10_J,axiom,
    ! [D2: int,D6: int,A4: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D6))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
         => ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2)))
           => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)),T2))) ) ) ) ).

% aset(10)
tff(fact_3137_aset_I9_J,axiom,
    ! [D2: int,D6: int,A4: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D6))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A4))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2)))
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)),T2))) ) ) ) ).

% aset(9)
tff(fact_3138_bset_I10_J,axiom,
    ! [D2: int,D6: int,B5: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D6))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
         => ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2)))
           => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6)),T2))) ) ) ) ).

% bset(10)
tff(fact_3139_bset_I9_J,axiom,
    ! [D2: int,D6: int,B5: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D6))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B5))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2)))
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D6)),T2))) ) ) ) ).

% bset(9)
tff(fact_3140_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(int,int,sgn_sgn(int),L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_3141_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_3142_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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_3143_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),U))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_3144_cos__45,axiom,
    aa(real,real,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_3145_sin__45,axiom,
    aa(real,real,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_3146_nat__dvd__iff,axiom,
    ! [Z: int,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,Z)),M2))
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z),aa(nat,int,semiring_1_of_nat(int),M2))) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => ( M2 = zero_zero(nat) ) ) ) ) ).

% nat_dvd_iff
tff(fact_3147_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_3148_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( 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))),N)) = 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),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_3149_ln__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% ln_sqrt
tff(fact_3150_cos__30,axiom,
    aa(real,real,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_3151_sin__60,axiom,
    aa(real,real,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_3152_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))))) ).

% arsinh_real_aux
tff(fact_3153_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,X)),N) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_3154_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_3155_powr__half__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),powr(real),X),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,X) ) ) ).

% powr_half_sqrt
tff(fact_3156_arsinh__real__def,axiom,
    ! [X: real] : aa(real,real,arsinh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))))) ).

% arsinh_real_def
tff(fact_3157_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(int,nat,nat2,K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)) ) ) ).

% even_nat_iff
tff(fact_3158_cos__x__y__le__one,axiom,
    ! [X: real,Y2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),one_one(real))) ).

% cos_x_y_le_one
tff(fact_3159_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),Y2)),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))))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_3160_cos__arctan,axiom,
    ! [X: real] : aa(real,real,cos(real),aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% cos_arctan
tff(fact_3161_sin__arctan,axiom,
    ! [X: real] : aa(real,real,sin(real),aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% sin_arctan
tff(fact_3162_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_3163_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sin(real),X)))
     => ( aa(real,real,sin(real),X) = 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(real,real,cos(real),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3164_arctan__half,axiom,
    ! [X: real] : aa(real,real,arctan,X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))) ).

% arctan_half
tff(fact_3165_arcosh__real__def,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( aa(real,real,arcosh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_3166_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa(real,real,cos(real),X) = zero_zero(real) )
    <=> ? [I2: int] :
          ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),I2))
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),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_3167_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa(real,real,sin(real),X) = zero_zero(real) )
    <=> ? [I2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),I2))
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),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_3168_cos__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( aa(real,real,cos(real),aa(real,real,arcsin,X)) = 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),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3169_sin__arccos__abs,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y2)),one_one(real)))
     => ( aa(real,real,sin(real),aa(real,real,arccos,Y2)) = 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),Y2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3170_sin__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( aa(real,real,sin(real),aa(real,real,arccos,X)) = 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),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).

% sin_arccos
tff(fact_3171_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),N)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_3172_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : 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),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_3173_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_3174_arcsin__0,axiom,
    aa(real,real,arcsin,zero_zero(real)) = zero_zero(real) ).

% arcsin_0
tff(fact_3175_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2584673776208193580ke_bit(A,zero_zero(nat),A3) = zero_zero(A) ) ).

% take_bit_0
tff(fact_3176_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_3177_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_3178_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3179_norm__cis,axiom,
    ! [A3: real] : real_V7770717601297561774m_norm(complex,cis(A3)) = one_one(real) ).

% norm_cis
tff(fact_3180_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : bit_concat_bit(N,K,zero_zero(int)) = bit_se2584673776208193580ke_bit(int,N,K) ).

% concat_bit_of_zero_2
tff(fact_3181_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_3182_cis__inverse,axiom,
    ! [A3: real] : aa(complex,complex,inverse_inverse(complex),cis(A3)) = cis(aa(real,real,uminus_uminus(real),A3)) ).

% cis_inverse
tff(fact_3183_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] :
          ( ( bit_se2584673776208193580ke_bit(A,N,one_one(A)) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_3184_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,bit_se2584673776208193580ke_bit(int,N,K))) = aa(int,A,ring_1_of_int(A),bit_se2584673776208193580ke_bit(int,N,K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_3185_arccos__minus__1,axiom,
    aa(real,real,arccos,aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_3186_cis__pi,axiom,
    cis(pi) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% cis_pi
tff(fact_3187_cos__arccos,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => ( aa(real,real,cos(real),aa(real,real,arccos,Y2)) = Y2 ) ) ) ).

% cos_arccos
tff(fact_3188_sin__arcsin,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => ( aa(real,real,sin(real),aa(real,real,arcsin,Y2)) = Y2 ) ) ) ).

% sin_arcsin
tff(fact_3189_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2584673776208193580ke_bit(A,N,A3)))
        <=> ( ( N = zero_zero(nat) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) ) ) ) ).

% even_take_bit_eq
tff(fact_3190_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: 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_3191_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_3192_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_3193_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_3194_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_3195_take__bit__minus,axiom,
    ! [N: nat,K: int] : bit_se2584673776208193580ke_bit(int,N,aa(int,int,uminus_uminus(int),bit_se2584673776208193580ke_bit(int,N,K))) = bit_se2584673776208193580ke_bit(int,N,aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_3196_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : bit_se2584673776208193580ke_bit(int,N,aa(int,int,aa(int,fun(int,int),minus_minus(int),bit_se2584673776208193580ke_bit(int,N,K)),bit_se2584673776208193580ke_bit(int,N,L))) = bit_se2584673776208193580ke_bit(int,N,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% take_bit_diff
tff(fact_3197_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M2: nat] : bit_se2584673776208193580ke_bit(A,N,aa(nat,A,semiring_1_of_nat(A),M2)) = aa(nat,A,semiring_1_of_nat(A),bit_se2584673776208193580ke_bit(nat,N,M2)) ) ).

% take_bit_of_nat
tff(fact_3198_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se2584673776208193580ke_bit(int,N,K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_3199_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se2584673776208193580ke_bit(int,N,K))) ).

% take_bit_nonnegative
tff(fact_3200_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2584673776208193580ke_bit(int,N,K)),zero_zero(int))) ).

% not_take_bit_negative
tff(fact_3201_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),bit_se2584673776208193580ke_bit(int,N,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% take_bit_int_greater_self_iff
tff(fact_3202_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A,B2: A] :
          ( ( bit_ri4674362597316999326ke_bit(A,N,A3) = bit_ri4674362597316999326ke_bit(A,N,B2) )
        <=> ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3) = bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_3203_cis__divide,axiom,
    ! [A3: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),cis(A3)),cis(B2)) = cis(aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),B2)) ).

% cis_divide
tff(fact_3204_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M2: nat,N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
         => ( bit_se2584673776208193580ke_bit(A,M2,bit_ri4674362597316999326ke_bit(A,N,A3)) = bit_se2584673776208193580ke_bit(A,M2,A3) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_3205_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,N,K) != zero_zero(int) )
     => ( bit_se2584673776208193580ke_bit(int,N,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),bit_se2584673776208193580ke_bit(int,N,K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_3206_arccos__le__arccos,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y2)),aa(real,real,arccos,X))) ) ) ) ).

% arccos_le_arccos
tff(fact_3207_arccos__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y2)),one_one(real))) )
     => ( ( aa(real,real,arccos,X) = aa(real,real,arccos,Y2) )
      <=> ( X = Y2 ) ) ) ).

% arccos_eq_iff
tff(fact_3208_arccos__le__mono,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y2)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,X)),aa(real,real,arccos,Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),X)) ) ) ) ).

% arccos_le_mono
tff(fact_3209_arcsin__le__arcsin,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y2))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3210_arcsin__minus,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_3211_arcsin__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y2)),one_one(real)))
       => ( ( aa(real,real,arcsin,X) = aa(real,real,arcsin,Y2) )
        <=> ( X = Y2 ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3212_arcsin__le__mono,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y2)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ) ) ).

% arcsin_le_mono
tff(fact_3213_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_bit0
tff(fact_3214_arccos__lbound,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y2))) ) ) ).

% arccos_lbound
tff(fact_3215_arccos__less__arccos,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,Y2)),aa(real,real,arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_3216_arccos__less__mono,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y2)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,X)),aa(real,real,arccos,Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),X)) ) ) ) ).

% arccos_less_mono
tff(fact_3217_arccos__ubound,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y2)),pi)) ) ) ).

% arccos_ubound
tff(fact_3218_arccos__cos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => ( aa(real,real,arccos,aa(real,real,cos(real),X)) = X ) ) ) ).

% arccos_cos
tff(fact_3219_arcsin__less__arcsin,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y2))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3220_arcsin__less__mono,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y2)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y2)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ) ) ).

% arcsin_less_mono
tff(fact_3221_cos__arccos__abs,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y2)),one_one(real)))
     => ( aa(real,real,cos(real),aa(real,real,arccos,Y2)) = Y2 ) ) ).

% cos_arccos_abs
tff(fact_3222_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Theta)),pi))
     => ( aa(real,real,arccos,aa(real,real,cos(real),Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_3223_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2584673776208193580ke_bit(int,N,K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% take_bit_int_less_exp
tff(fact_3224_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( bit_se2584673776208193580ke_bit(A,N,A3) = zero_zero(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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))),N)),A3)) ) ) ).

% take_bit_eq_0_iff
tff(fact_3225_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N),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),bit_se2584673776208193580ke_bit(int,N,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_3226_arccos__lt__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,arccos,Y2)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,Y2)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_3227_arccos__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y2)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y2)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_3228_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),bit_se2584673776208193580ke_bit(int,N,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_3229_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2584673776208193580ke_bit(int,N,K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_3230_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => ( aa(real,real,sin(real),aa(real,real,arccos,X)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3231_arccos__cos2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),X))
       => ( aa(real,real,arccos,aa(real,real,cos(real),X)) = aa(real,real,uminus_uminus(real),X) ) ) ) ).

% arccos_cos2
tff(fact_3232_arccos__minus,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_3233_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => ( aa(real,real,cos(real),aa(real,real,arcsin,X)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3234_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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))),N)))
       => ( bit_se2584673776208193580ke_bit(int,N,K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_3235_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,N,K) = K )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),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))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_3236_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : 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),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_3237_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,N,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))),N)),one_one(int)) )
     => ( bit_se2584673776208193580ke_bit(int,N,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),bit_se2584673776208193580ke_bit(int,N,K)) ) ) ).

% take_bit_incr_eq
tff(fact_3238_arccos,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y2)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y2)),pi))
          & ( aa(real,real,cos(real),aa(real,real,arccos,Y2)) = Y2 ) ) ) ) ).

% arccos
tff(fact_3239_arccos__minus__abs,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_3240_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),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,N))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_3241_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,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_3242_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : 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_3243_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,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_3244_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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))),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se2584673776208193580ke_bit(int,N,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))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_3245_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),bit_se2584673776208193580ke_bit(int,N,K))) ) ).

% take_bit_int_greater_eq
tff(fact_3246_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : bit_ri4674362597316999326ke_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_3247_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: 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 )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
             => ( bit_se2584673776208193580ke_bit(A,N,A3) = zero_zero(A) ) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
             => ( bit_se2584673776208193580ke_bit(A,N,A3) = 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))),N)),one_one(A)) ) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_3248_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : 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),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_3249_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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))),N)))
       => ( bit_se2584673776208193580ke_bit(int,N,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))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_3250_arccos__le__pi2,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y2)),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_3251_arcsin__lt__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y2)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,Y2)),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_3252_arcsin__bounded,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y2)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y2)),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_3253_arcsin__ubound,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y2)),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_3254_arcsin__lbound,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y2))) ) ) ).

% arcsin_lbound
tff(fact_3255_arcsin__sin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( aa(real,real,arcsin,aa(real,real,sin(real),X)) = X ) ) ) ).

% arcsin_sin
tff(fact_3256_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,N,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_3257_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( bit_se2584673776208193580ke_bit(A,N,A3) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( bit_se2584673776208193580ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),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_3258_le__arcsin__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),aa(real,real,arcsin,X)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sin(real),Y2)),X)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3259_arcsin__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,X)),Y2))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,sin(real),Y2))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3260_arcsin__pi,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y2)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y2)),pi))
          & ( aa(real,real,sin(real),aa(real,real,arcsin,Y2)) = Y2 ) ) ) ) ).

% arcsin_pi
tff(fact_3261_arcsin,axiom,
    ! [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y2)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y2)),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,sin(real),aa(real,real,arcsin,Y2)) = Y2 ) ) ) ) ).

% arcsin
tff(fact_3262_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa(real,real,arccos,aa(real,real,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),K2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_3263_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_3264_exp__two__pi__i_H,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),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_3265_exp__two__pi__i,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),aa(num,num,bit0,one2))),real_Vector_of_real(complex,pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_3266_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(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),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_3267_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),nat_set_decode(X)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X)) ) ).

% set_decode_0
tff(fact_3268_nat__of__bool,axiom,
    ! [P: bool] : aa(int,nat,nat2,aa(bool,int,zero_neq_one_of_bool(int),P)) = aa(bool,nat,zero_neq_one_of_bool(nat),P) ).

% nat_of_bool
tff(fact_3269_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( pp(P)
           => pp(Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_3270_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = zero_zero(A) )
        <=> ~ pp(P) ) ) ).

% of_bool_eq_0_iff
tff(fact_3271_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fFalse) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_3272_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( ~ pp(P)
            & pp(Q) ) ) ) ).

% of_bool_less_iff
tff(fact_3273_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = one_one(A) )
        <=> pp(P) ) ) ).

% of_bool_eq_1_iff
tff(fact_3274_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fTrue) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_3275_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: bool] : aa(nat,A,semiring_1_of_nat(A),aa(bool,nat,zero_neq_one_of_bool(nat),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% of_nat_of_bool
tff(fact_3276_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: bool] : aa(A,A,abs_abs(A),aa(bool,A,zero_neq_one_of_bool(A),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% abs_bool_eq
tff(fact_3277_of__int__of__bool,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: bool] : aa(int,A,ring_1_of_int(A),aa(bool,int,zero_neq_one_of_bool(int),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% of_int_of_bool
tff(fact_3278_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P)))
        <=> pp(P) ) ) ).

% zero_less_of_bool_iff
tff(fact_3279_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A)))
        <=> ~ pp(P) ) ) ).

% of_bool_less_one_iff
tff(fact_3280_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: bool] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,P)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(bool,A,zero_neq_one_of_bool(A),P)) ) ).

% of_bool_not_iff
tff(fact_3281_Suc__0__mod__eq,axiom,
    ! [N: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),aa(nat,nat,suc,zero_zero(nat))))) ).

% Suc_0_mod_eq
tff(fact_3282_norm__ii,axiom,
    real_V7770717601297561774m_norm(complex,imaginary_unit) = one_one(real) ).

% norm_ii
tff(fact_3283_complex__i__mult__minus,axiom,
    ! [X: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),X)) = aa(complex,complex,uminus_uminus(complex),X) ).

% complex_i_mult_minus
tff(fact_3284_inverse__i,axiom,
    aa(complex,complex,inverse_inverse(complex),imaginary_unit) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% inverse_i
tff(fact_3285_sgn__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,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).

% sgn_mult_self_eq
tff(fact_3286_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).

% sgn_abs
tff(fact_3287_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_3288_take__bit__of__Suc__0,axiom,
    ! [N: nat] : bit_se2584673776208193580ke_bit(nat,N,aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_3289_divide__i,axiom,
    ! [X: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,uminus_uminus(complex),imaginary_unit)),X) ).

% divide_i
tff(fact_3290_i__squared,axiom,
    aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),imaginary_unit) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% i_squared
tff(fact_3291_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_3292_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% sgn_of_nat
tff(fact_3293_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: bool] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(bool,A,zero_neq_one_of_bool(A),B2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_3294_divide__numeral__i,axiom,
    ! [Z: complex,N: num] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),N)),imaginary_unit)) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z))),aa(num,complex,numeral_numeral(complex),N)) ).

% divide_numeral_i
tff(fact_3295_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,N)),nat_set_decode(X)))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% set_decode_Suc
tff(fact_3296_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_3297_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_3298_i__even__power,axiom,
    ! [N: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),imaginary_unit),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(num,nat,numeral_numeral(nat),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))),N) ).

% i_even_power
tff(fact_3299_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% one_div_2_pow_eq
tff(fact_3300_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% bits_1_div_exp
tff(fact_3301_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: nat,N: nat] : bit_se2584673776208193580ke_bit(A,M2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% take_bit_of_exp
tff(fact_3302_exp__pi__i_H,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,pi))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i'
tff(fact_3303_exp__pi__i,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,pi)),imaginary_unit)) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i
tff(fact_3304_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_3305_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool,Q5: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P2) = aa(bool,A,zero_neq_one_of_bool(A),Q5) )
        <=> ( pp(P2)
          <=> pp(Q5) ) ) ) ).

% of_bool_eq_iff
tff(fact_3306_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fconj(P,Q)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_conj
tff(fact_3307_complex__i__not__one,axiom,
    imaginary_unit != one_one(complex) ).

% complex_i_not_one
tff(fact_3308_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P))) ) ).

% zero_less_eq_of_bool
tff(fact_3309_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A))) ) ).

% of_bool_less_eq_one
tff(fact_3310_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ~ ( ( pp(P2)
                & ~ pp(aa(A,bool,P,one_one(A))) )
              | ( ~ pp(P2)
                & ~ pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool_asm
tff(fact_3311_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ( ( pp(P2)
             => pp(aa(A,bool,P,one_one(A))) )
            & ( ~ pp(P2)
             => pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool
tff(fact_3312_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool] :
          ( ( pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = one_one(A) ) )
          & ( ~ pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = zero_zero(A) ) ) ) ) ).

% of_bool_def
tff(fact_3313_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),W) = Z )
    <=> ( W = aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_3314_complex__i__not__neg__numeral,axiom,
    ! [W: num] : imaginary_unit != aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) ).

% complex_i_not_neg_numeral
tff(fact_3315_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_3316_Complex__eq__i,axiom,
    ! [X: real,Y2: real] :
      ( ( complex2(X,Y2) = imaginary_unit )
    <=> ( ( X = zero_zero(real) )
        & ( Y2 = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_3317_i__mult__Complex,axiom,
    ! [A3: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),complex2(A3,B2)) = complex2(aa(real,real,uminus_uminus(real),B2),A3) ).

% i_mult_Complex
tff(fact_3318_Complex__mult__i,axiom,
    ! [A3: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A3,B2)),imaginary_unit) = complex2(aa(real,real,uminus_uminus(real),B2),A3) ).

% Complex_mult_i
tff(fact_3319_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( bit_se2584673776208193580ke_bit(nat,N,aa(int,nat,nat2,K)) = aa(int,nat,nat2,bit_se2584673776208193580ke_bit(int,N,K)) ) ) ).

% take_bit_nat_eq
tff(fact_3320_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( aa(int,nat,nat2,bit_se2584673776208193580ke_bit(int,N,K)) = bit_se2584673776208193580ke_bit(nat,N,aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_3321_subset__decode__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),nat_set_decode(M2)),nat_set_decode(N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% subset_decode_imp_le
tff(fact_3322_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( bit_se2584673776208193580ke_bit(nat,N,M2) = M2 )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_3323_take__bit__nat__less__exp,axiom,
    ! [N: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),bit_se2584673776208193580ke_bit(nat,N,M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% take_bit_nat_less_exp
tff(fact_3324_take__bit__nat__eq__self,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N)))
     => ( bit_se2584673776208193580ke_bit(nat,N,M2) = M2 ) ) ).

% take_bit_nat_eq_self
tff(fact_3325_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),bit_se2584673776208193580ke_bit(nat,N,M2)),M2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N)),M2)) ) ).

% take_bit_nat_less_self_iff
tff(fact_3326_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,bool),A3: A] :
          ( ! [A6: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A6),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A6 )
             => pp(aa(A,bool,P,A6)) )
         => ( ! [A6: A,B4: bool] :
                ( pp(aa(A,bool,P,A6))
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A6))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A6 )
                 => pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A6)))) ) )
           => pp(aa(A,bool,P,A3)) ) ) ) ).

% bits_induct
tff(fact_3327_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M2: nat,N: 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))),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)) ) ).

% exp_mod_exp
tff(fact_3328_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_3329_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)))),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),M2),N))) ) ).

% exp_div_exp_eq
tff(fact_3330_cmod__unit__one,axiom,
    ! [A3: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,aa(real,real,cos(real),A3))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,aa(real,real,sin(real),A3))))) = one_one(real) ).

% cmod_unit_one
tff(fact_3331_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M2: nat] :
      ( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(int) ) )
      & ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
            | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)) ) )
          & ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_3332_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero(int) )
       => ( modulo_modulo(int,K,L) = K ) )
      & ( ( L != zero_zero(int) )
       => ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))) ) )
          & ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),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_3333_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M2: nat] :
      ( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M2)) ) )
      & ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
            | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M2,N))) ) )
          & ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2)))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M2,N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_3334_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero(int) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) )
      & ( ( L != zero_zero(int) )
       => ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(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,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,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(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_3335_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_3336_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_3337_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_3338_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(arg(Z)) )
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z)))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),arg(Z)),pi)) ) ) ).

% Arg_correct
tff(fact_3339_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(X) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
         => ( arg(Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_3340_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% csqrt_eq_1
tff(fact_3341_csqrt__1,axiom,
    csqrt(one_one(complex)) = one_one(complex) ).

% csqrt_1
tff(fact_3342_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),arg(Z)),pi)) ) ).

% Arg_bounded
tff(fact_3343_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_3344_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : bit_ri4674362597316999326ke_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N))),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_3345_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_3346_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_3347_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A6: bool,B4: bool] : A1 = vEBT_Leaf(A6,B4)
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                     => vEBT_invar_vebt(X3,N2) )
                 => ( vEBT_invar_vebt(Summary2,M3)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M3) )
                     => ( ( M3 = N2 )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M3) )
                         => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_12))
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                 => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_12)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_invar_vebt(X3,N2) )
                   => ( vEBT_invar_vebt(Summary2,M3)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M3) )
                       => ( ( M3 = aa(nat,nat,suc,N2) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M3) )
                           => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_12))
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                   => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_12)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi: 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),product_Pair(nat,nat,Mi),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_invar_vebt(X3,N2) )
                     => ( vEBT_invar_vebt(Summary2,M3)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M3) )
                         => ( ( M3 = N2 )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M3) )
                             => ( ! [I4: nat] :
                                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),M3)))
                                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13))
                                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                               => ( ( ( Mi = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                       => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma2))
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),Deg2)))
                                     => ~ ( ( Mi != Ma2 )
                                         => ! [I4: nat] :
                                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),M3)))
                                             => ( ( ( vEBT_VEBT_high(Ma2,N2) = I4 )
                                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N2))) )
                                                & ! [X3: nat] :
                                                    ( ( ( vEBT_VEBT_high(X3,N2) = I4 )
                                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X3,N2))) )
                                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X3))
                                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi: 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),product_Pair(nat,nat,Mi),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                           => vEBT_invar_vebt(X3,N2) )
                       => ( vEBT_invar_vebt(Summary2,M3)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M3) )
                           => ( ( M3 = aa(nat,nat,suc,N2) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M3) )
                               => ( ! [I4: nat] :
                                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),M3)))
                                     => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13))
                                      <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                                 => ( ( ( Mi = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                         => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma2))
                                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),Deg2)))
                                       => ~ ( ( Mi != Ma2 )
                                           => ! [I4: nat] :
                                                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),M3)))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N2) = I4 )
                                                   => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N2))) )
                                                  & ! [X3: nat] :
                                                      ( ( ( vEBT_VEBT_high(X3,N2) = I4 )
                                                        & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X3,N2))) )
                                                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X3))
                                                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3348_option_Oinject,axiom,
    ! [A: $tType,X23: A,Y23: A] :
      ( ( aa(A,option(A),some(A),X23) = aa(A,option(A),some(A),Y23) )
    <=> ( X23 = Y23 ) ) ).

% option.inject
tff(fact_3349_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% mask_nat_positive_iff
tff(fact_3350_not__Some__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ! [Y4: A] : X != aa(A,option(A),some(A),Y4)
    <=> ( X = none(A) ) ) ).

% not_Some_eq
tff(fact_3351_not__None__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ( X != none(A) )
    <=> ? [Y4: A] : X = aa(A,option(A),some(A),Y4) ) ).

% not_None_eq
tff(fact_3352_mi__ma__2__deg,axiom,
    ! [Mi2: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi2),Ma))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),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_2_deg
tff(fact_3353_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_3354_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( ( bit_se2239418461657761734s_mask(A,N) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_3355_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_3356_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi2: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),Deg))
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Deg,TreeList,Summary)),X))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
          | ( X = Mi2 )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_3357_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_3358_member__inv,axiom,
    ! [Mi2: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Deg,TreeList,Summary)),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
        & ( ( X = Mi2 )
          | ( X = Ma )
          | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
            & pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% member_inv
tff(fact_3359_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,M2))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M2)),N)) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_3360_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M2))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M2)),N)) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_3361_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi2: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),Deg))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
         => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_3362_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_3363_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_3364_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_ri4674362597316999326ke_bit(int,N,K)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_3365_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_ri4674362597316999326ke_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_3366_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W)))),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_3367_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(nat,nat,suc,N)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_3368_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),zero_zero(nat)))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) ) ) ).

% bit_0
tff(fact_3369_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W)))),aa(num,nat,numeral_numeral(nat),N)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),pred_numeral(N))) ) ).

% bit_minus_numeral_int(1)
tff(fact_3370_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(num,nat,numeral_numeral(nat),N)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),pred_numeral(N))) ) ).

% bit_minus_numeral_int(2)
tff(fact_3371_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N))
        <=> ( ( N = zero_zero(nat) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) ) ) ) ).

% bit_mod_2_iff
tff(fact_3372_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se2239418461657761734s_mask(nat,N)) = bit_se2239418461657761734s_mask(A,N) ) ).

% of_nat_mask_eq
tff(fact_3373_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M2)),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M2),N)) ) ) ).

% bit_of_nat_iff_bit
tff(fact_3374_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod(nat,nat)] :
      ~ ! [K2: nat,M3: nat] : X != aa(nat,product_prod(nat,nat),product_Pair(nat,nat,K2),M3) ).

% prod_decode_aux.cases
tff(fact_3375_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X23: A] : none(A) != aa(A,option(A),some(A),X23) ).

% option.distinct(1)
tff(fact_3376_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X23: A] :
      ( ( Option = aa(A,option(A),some(A),X23) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_3377_option_Oexhaust,axiom,
    ! [A: $tType,Y2: option(A)] :
      ( ( Y2 != none(A) )
     => ~ ! [X24: A] : Y2 != aa(A,option(A),some(A),X24) ) ).

% option.exhaust
tff(fact_3378_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ? [X_13: option(A)] : pp(aa(option(A),bool,P,X_13))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        | ? [X4: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_ex
tff(fact_3379_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ! [X_13: option(A)] : pp(aa(option(A),bool,P,X_13))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        & ! [X4: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_all
tff(fact_3380_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,X: option(A),P: fun(option(A),fun(option(B),bool)),Y2: option(B)] :
      ( ( ( X = none(A) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y2)) )
     => ( ( ( Y2 = none(B) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y2)) )
       => ( ! [A6: A,B4: B] :
              ( ( X = aa(A,option(A),some(A),A6) )
             => ( ( Y2 = aa(B,option(B),some(B),B4) )
               => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y2)) ) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y2)) ) ) ) ).

% combine_options_cases
tff(fact_3381_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),bot_bot(A))) ) ).

% bot.extremum_strict
tff(fact_3382_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( ( A3 != bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),A3)) ) ) ).

% bot.not_eq_extremum
tff(fact_3383_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi2: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),zero_zero(nat),Va2,Vb),X)
    <=> ( ( X = Mi2 )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3384_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,N))) ) ).

% not_bit_1_Suc
tff(fact_3385_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),N))
        <=> ( N = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_3386_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% bit_numeral_simps(1)
tff(fact_3387_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) ) ) ).

% diff_shunt_var
tff(fact_3388_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se2584673776208193580ke_bit(A,M2,A3)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ) ).

% bit_take_bit_iff
tff(fact_3389_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: bool,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(bool,A,zero_neq_one_of_bool(A),B2)),N))
        <=> ( pp(B2)
            & ( N = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_3390_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,N))) ).

% mask_nonnegative_int
tff(fact_3391_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2239418461657761734s_mask(int,N)),zero_zero(int))) ).

% not_mask_negative_int
tff(fact_3392_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va2: 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),Va2,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_3393_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,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))),N))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% bit_not_int_iff'
tff(fact_3394_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_3395_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_3396_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),bit_se2239418461657761734s_mask(nat,N))) ) ).

% less_mask
tff(fact_3397_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(X)
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3398_option_Osize_I4_J,axiom,
    ! [A: $tType,X23: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X23)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_3399_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M2: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),bit_se2584673776208193580ke_bit(int,M2,K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3400_bit__concat__bit__iff,axiom,
    ! [M2: nat,K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,bit_concat_bit(M2,K,L)),N))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) )
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_3401_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_3402_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : bit_ri4674362597316999326ke_bit(int,N,K) = bit_concat_bit(N,K,aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_3403_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X23: A] : aa(option(A),nat,size_option(A,X),aa(A,option(A),some(A),X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_3404_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: 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))),N) = zero_zero(A) )
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_3405_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,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)))),N)) ) ) ).

% bit_Suc
tff(fact_3406_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: 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 )
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
          <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_3407_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N2))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),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_3408_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y2: bool] :
      ( ( vEBT_VEBT_minNull(X)
      <=> pp(Y2) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y2) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y2) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y2) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy)
               => ~ pp(Y2) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => pp(Y2) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3409_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,N,K) = bit_se2239418461657761734s_mask(int,N) )
    <=> ( bit_se2584673776208193580ke_bit(int,N,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_3410_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),M))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) )
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))))
              <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_3411_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_3412_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),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))),N)))) ) ) ).

% bit_iff_odd
tff(fact_3413_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% Suc_mask_eq_exp
tff(fact_3414_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),bit_se2239418461657761734s_mask(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% mask_nat_less_exp
tff(fact_3415_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
    <=> ~ pp(aa(int,bool,aa(int,fun(int,bool),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))),N)))) ) ).

% bit_int_def
tff(fact_3416_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,N)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_3417_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_3418_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_3419_mask__nat__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(nat,N) = 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))),N)),one_one(nat)) ).

% mask_nat_def
tff(fact_3420_mask__half__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ).

% mask_half_int
tff(fact_3421_mask__int__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(int,N) = 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))),N)),one_one(int)) ).

% mask_int_def
tff(fact_3422_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,N: nat] :
          ( ! [J: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,J)))
         => ( pp(aa(nat,bool,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))),N))
          <=> ( ( ( N = zero_zero(nat) )
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) )
              & ( ( N != zero_zero(nat) )
               => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)),N)) ) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3423_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
        <=> ( ( ( N = zero_zero(nat) )
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) )
            & ( ( N != zero_zero(nat) )
             => pp(aa(nat,bool,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),N),one_one(nat)))) ) ) ) ) ).

% bit_rec
tff(fact_3424_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,N) = 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))),N)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_3425_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X2,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] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_13))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi2 = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_1)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi2),Ma))
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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)))
                     => ( ( ( Mi2 != Ma )
                         => ! [I3: nat] :
                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X2: nat] :
                                    ( ( ( vEBT_VEBT_high(X2,N) = I3 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(X2,N))) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X2))
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_3426_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X2,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] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_13))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi2 = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_1)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi2),Ma))
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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)))
                     => ( ( ( Mi2 != Ma )
                         => ! [I3: nat] :
                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X2: nat] :
                                    ( ( ( vEBT_VEBT_high(X2,N) = I3 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(X2,N))) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X2))
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_3427_unset__bit__eq,axiom,
    ! [N: nat,K: int] : bit_se2638667681897837118et_bit(int,N,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(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% unset_bit_eq
tff(fact_3428_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))),bit_se2584673776208193580ke_bit(int,N,K)) ).

% take_bit_Suc_from_most
tff(fact_3429_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,N,K) = bit_se2239418461657761734s_mask(int,N) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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))),N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_3430_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : size_num(aa(num,num,bit0,X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_3431_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A5: bool,B3: bool] : A1 = vEBT_Leaf(A5,B3)
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary3,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N5)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary3,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ! [I2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N5)))
               => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),X_13))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I2)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi3),Ma3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N5)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N5) = I2 )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(Ma3,N5))) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N5) = I2 )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(X4,N5))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X4))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N5)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ! [I2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),aa(nat,nat,suc,N5))))
               => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),X_13))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I2)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi3),Ma3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),aa(nat,nat,suc,N5))))
                 => ( ( ( vEBT_VEBT_high(Ma3,N5) = I2 )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(Ma3,N5))) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N5) = I2 )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(X4,N5))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X4))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3432_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,R: A,Q5: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),product_Pair(A,A,Q5),R)) = aa(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))),Q5)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(num,A,numeral_numeral(A),L))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),product_Pair(A,A,Q5),R)) = aa(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))),Q5)),R) ) ) ) ) ).

% divmod_step_eq
tff(fact_3433_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5)) ) ).

% Diff_eq_empty_iff
tff(fact_3434_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q5: A,R: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),product_Pair(A,A,Q5),R))
        <=> ( R = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_3435_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2))))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys2)),N) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(list(B),nat,size_size(list(B)),Ys2)))),aa(nat,B,nth(B,Ys2),modulo_modulo(nat,N,aa(list(B),nat,size_size(list(B)),Ys2)))) ) ) ).

% product_nth
tff(fact_3436_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero(int) )
          | ( L = zero_zero(int) ) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = zero_zero(int) ) )
      & ( ~ ( ( K = zero_zero(int) )
            | ( L = zero_zero(int) ) )
       => ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = L ) )
          & ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = K ) )
              & ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),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_3437_Compl__anti__mono,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B5)),aa(set(A),set(A),uminus_uminus(set(A)),A4))) ) ).

% Compl_anti_mono
tff(fact_3438_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B5)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A4)) ) ).

% Compl_subset_Compl_iff
tff(fact_3439_Diff__idemp,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) ).

% Diff_idemp
tff(fact_3440_Diff__iff,axiom,
    ! [A: $tType,C2: A,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A4))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B5)) ) ) ).

% Diff_iff
tff(fact_3441_DiffI,axiom,
    ! [A: $tType,C2: A,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A4))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B5))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5))) ) ) ).

% DiffI
tff(fact_3442_Diff__cancel,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),A4) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_3443_empty__Diff,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A4) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_3444_Diff__empty,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),bot_bot(set(A))) = A4 ).

% Diff_empty
tff(fact_3445_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_3446_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),X) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_3447_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_3448_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_3449_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,bool)) ) ) ).

% bit_0_eq
tff(fact_3450_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_3451_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_3452_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = X ) ).

% bit.conj_one_right
tff(fact_3453_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_3454_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% and_negative_int_iff
tff(fact_3455_length__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys2)) = 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)),Ys2)) ).

% length_product
tff(fact_3456_and__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y2: 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,Y2))) = one_one(A) ) ).

% and_numerals(2)
tff(fact_3457_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_3458_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_3459_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y2: 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,Y2))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_3460_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_3461_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_3462_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_3463_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_3464_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y2: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y2))) = 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),X)),aa(num,A,numeral_numeral(A),Y2)))) ) ).

% and_numerals(7)
tff(fact_3465_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_3466_DiffD2,axiom,
    ! [A: $tType,C2: A,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B5)) ) ).

% DiffD2
tff(fact_3467_DiffD1,axiom,
    ! [A: $tType,C2: A,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A4)) ) ).

% DiffD1
tff(fact_3468_DiffE,axiom,
    ! [A: $tType,C2: A,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A4))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B5)) ) ) ).

% DiffE
tff(fact_3469_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_and_eq
tff(fact_3470_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_3471_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),N))
    <=> ( N = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_3472_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_3473_AND__upper2_H,axiom,
    ! [Y2: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y2),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y2)),Z)) ) ) ).

% AND_upper2'
tff(fact_3474_AND__upper1_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y2),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y2),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_3475_AND__upper2,axiom,
    ! [Y2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y2)),Y2)) ) ).

% AND_upper2
tff(fact_3476_AND__upper1,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y2)),X)) ) ).

% AND_upper1
tff(fact_3477_AND__lower,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y2))) ) ).

% AND_lower
tff(fact_3478_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,D5: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(Uu2,Uv2)),D5)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Deg3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_3479_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),N))) ).

% not_bit_Suc_0_numeral
tff(fact_3480_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K)) ) ).

% and_less_eq
tff(fact_3481_AND__upper1_H_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y2),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y2),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_3482_AND__upper2_H_H,axiom,
    ! [Y2: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y2),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y2)),Z)) ) ) ).

% AND_upper2''
tff(fact_3483_Diff__mono,axiom,
    ! [A: $tType,A4: set(A),C5: set(A),D6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D6),B5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),D6))) ) ) ).

% Diff_mono
tff(fact_3484_Diff__subset,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),A4)) ).

% Diff_subset
tff(fact_3485_double__diff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A),C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C5))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A4)) = A4 ) ) ) ).

% double_diff
tff(fact_3486_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B5))
     => ? [B4: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4))) ) ).

% psubset_imp_ex_mem
tff(fact_3487_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(int,nat,nat2,K)),N))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ) ).

% bit_nat_iff
tff(fact_3488_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A6: bool,B4: bool,X2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A6,B4)),X2)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Ux2)
       => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT,X2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2)),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3489_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_3490_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_3491_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: 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))),N)) = zero_zero(A) )
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_3492_bit__nat__def,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M2),N))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),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),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ) ).

% bit_nat_def
tff(fact_3493_subset__Compl__self__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4)))
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_3494_vebt__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A6: bool,B4: bool,X2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A6,B4)),X2)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),X2)
       => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT,X2: nat] : X != aa(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)),V4),zero_zero(nat),Uy,Uz2)),X2)
         => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X2: nat] : X != aa(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)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X2)
           => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X2: nat] : X != aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X2) ) ) ) ) ).

% vebt_member.cases
tff(fact_3495_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(Uu2,Uv2)),Uw2)
     => ( ! [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT,Uz2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)),Uz2)
       => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X2: nat] : X != aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),X2)
         => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X2: nat] : X != aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),X2)
           => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT,X2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd)),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3496_and__int__rec,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),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_3497_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M2),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,M2))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M2),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit0,M2),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_3498_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M2),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M2))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M2),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M2),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_3499_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q5: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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),product_Pair(int,int,Q5),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),product_Pair(int,int,Q5),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_3500_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y2 )
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),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))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),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)))))) )
         => ( Y2 = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),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))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),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)))))) )
         => ( Y2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa))))),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),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% and_int.elims
tff(fact_3501_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),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))))))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),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,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))) ) )
      & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),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))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),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,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),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_3502_ComplI,axiom,
    ! [A: $tType,C2: A,A4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A4))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),uminus_uminus(set(A)),A4))) ) ).

% ComplI
tff(fact_3503_Compl__iff,axiom,
    ! [A: $tType,C2: A,A4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),uminus_uminus(set(A)),A4)))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A4)) ) ).

% Compl_iff
tff(fact_3504_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(set(A),set(A),uminus_uminus(set(A)),B5) )
    <=> ( A4 = B5 ) ) ).

% Compl_eq_Compl_iff
tff(fact_3505_Diff__insert0,axiom,
    ! [A: $tType,X: A,A4: set(A),B5: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
     => ( 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,X),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) ) ) ).

% Diff_insert0
tff(fact_3506_insert__Diff1,axiom,
    ! [A: $tType,X: A,B5: set(A),A4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,X),A4)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) ) ) ).

% insert_Diff1
tff(fact_3507_insert__Diff__single,axiom,
    ! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),insert(A,A3),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))))) = aa(set(A),set(A),insert(A,A3),A4) ).

% insert_Diff_single
tff(fact_3508_subset__Compl__singleton,axiom,
    ! [A: $tType,A4: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A4)) ) ).

% subset_Compl_singleton
tff(fact_3509_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num] : unique8689654367752047608divmod(A,M2,one2) = aa(A,product_prod(A,A),product_Pair(A,A,aa(num,A,numeral_numeral(A),M2)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_3510_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_3511_and__nat__numerals_I1_J,axiom,
    ! [Y2: 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,Y2))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_3512_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit0,N)) = aa(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_3513_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N)) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_3514_and__nat__numerals_I2_J,axiom,
    ! [Y2: 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,Y2))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_3515_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_3516_Suc__0__and__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),N) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Suc_0_and_eq
tff(fact_3517_and__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),N),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% and_Suc_0_eq
tff(fact_3518_insert__Diff__if,axiom,
    ! [A: $tType,X: A,B5: set(A),A4: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B5))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,X),A4)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B5))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,X),A4)),B5) = aa(set(A),set(A),insert(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) ) ) ) ).

% insert_Diff_if
tff(fact_3519_ComplD,axiom,
    ! [A: $tType,C2: A,A4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),uminus_uminus(set(A)),A4)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A4)) ) ).

% ComplD
tff(fact_3520_double__complement,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = A4 ).

% double_complement
tff(fact_3521_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_3522_div__int__unique,axiom,
    ! [K: int,L: int,Q5: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q5),R))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q5 ) ) ).

% div_int_unique
tff(fact_3523_Diff__insert__absorb,axiom,
    ! [A: $tType,X: A,A4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,X),A4)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_3524_Diff__insert2,axiom,
    ! [A: $tType,A4: set(A),A3: A,B5: 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,A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))),B5) ).

% Diff_insert2
tff(fact_3525_insert__Diff,axiom,
    ! [A: $tType,A3: A,A4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
     => ( aa(set(A),set(A),insert(A,A3),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))))) = A4 ) ) ).

% insert_Diff
tff(fact_3526_Diff__insert,axiom,
    ! [A: $tType,A4: set(A),A3: A,B5: 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,A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_3527_Compl__insert,axiom,
    ! [A: $tType,X: A,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,X),A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_3528_subset__Diff__insert,axiom,
    ! [A: $tType,A4: set(A),B5: set(A),X: A,C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),aa(set(A),set(A),insert(A,X),C5))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C5)))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4)) ) ) ).

% subset_Diff_insert
tff(fact_3529_and__nat__def,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M2),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% and_nat_def
tff(fact_3530_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q5: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q5),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_3531_Diff__single__insert,axiom,
    ! [A: $tType,A4: set(A),X: A,B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(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,X),bot_bot(set(A))))),B5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),insert(A,X),B5))) ) ).

% Diff_single_insert
tff(fact_3532_subset__insert__iff,axiom,
    ! [A: $tType,A4: set(A),X: A,B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),insert(A,X),B5)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(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,X),bot_bot(set(A))))),B5)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5)) ) ) ) ).

% subset_insert_iff
tff(fact_3533_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(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_3534_divmod__int__def,axiom,
    ! [M2: num,N: num] : unique8689654367752047608divmod(int,M2,N) = aa(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),M2)),aa(num,int,numeral_numeral(int),N))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M2),aa(num,int,numeral_numeral(int),N))) ).

% divmod_int_def
tff(fact_3535_psubset__insert__iff,axiom,
    ! [A: $tType,A4: set(A),X: A,B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),aa(set(A),set(A),insert(A,X),B5)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B5)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B5))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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,X),bot_bot(set(A))))),B5)) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_3536_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)))
     => ( set_or1337092689740270186AtMost(int,M2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = aa(set(int),set(int),insert(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),set_or1337092689740270186AtMost(int,M2,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_3537_simp__from__to,axiom,
    ! [J2: int,I: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I))
       => ( set_or1337092689740270186AtMost(int,I,J2) = bot_bot(set(int)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I))
       => ( set_or1337092689740270186AtMost(int,I,J2) = 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)),J2)) ) ) ) ).

% simp_from_to
tff(fact_3538_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : unique8689654367752047608divmod(A,M2,N) = aa(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),M2)),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_3539_divmod_H__nat__def,axiom,
    ! [M2: num,N: num] : unique8689654367752047608divmod(nat,M2,N) = aa(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),M2)),aa(num,nat,numeral_numeral(nat),N))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N))) ).

% divmod'_nat_def
tff(fact_3540_zminus1__lemma,axiom,
    ! [A3: int,B2: int,Q5: int,R: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q5),R))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B2,aa(int,product_prod(int,int),product_Pair(int,int,if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int)),aa(int,int,uminus_uminus(int),Q5),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q5)),one_one(int)))),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R)))) ) ) ).

% zminus1_lemma
tff(fact_3541_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q5: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q5),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),Q5)),R) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),L)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),R))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int))) ) )
            & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( Q5 = zero_zero(int) ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_3542_and__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ( M2 = zero_zero(nat) )
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M2),N) = zero_zero(nat) ) )
      & ( ~ ( ( M2 = zero_zero(nat) )
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,N,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),M2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_nat_unfold
tff(fact_3543_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q5: int] :
      ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),L) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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),Q5),L)),R) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q5),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_3544_and__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fconj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2)),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))))),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),M2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% and_nat_rec
tff(fact_3545_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),K3) ) )
        | ? [L3: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q4),zero_zero(int)) )
            & ( L3 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q4),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L3) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L3)))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_3546_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),product_Pair(int,int,zero_zero(int)),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(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),product_Pair(int,int,Q3),R3) )
               => ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),A22) )
                 => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),A22)))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_3547_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
           => ( unique8689654367752047608divmod(A,M2,N) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),M2)) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
           => ( unique8689654367752047608divmod(A,M2,N) = unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M2,aa(num,num,bit0,N))) ) ) ) ) ).

% divmod_divmod_step
tff(fact_3548_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q5: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q5),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),product_Pair(int,int,Q5),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_3549_minus__one__div__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_div_numeral
tff(fact_3550_one__div__minus__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% one_div_minus_numeral
tff(fact_3551_Divides_Oadjust__div__eq,axiom,
    ! [Q5: int,R: int] : adjust_div(aa(int,product_prod(int,int),product_Pair(int,int,Q5),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q5),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int))))) ).

% Divides.adjust_div_eq
tff(fact_3552_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,K),L))
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),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))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),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,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),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))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),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,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),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_3553_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y2 )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,X),Xa))
       => ~ ( ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),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))))))
                  & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),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)))))) )
               => ( Y2 = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa))))) ) )
              & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),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))))))
                    & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),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)))))) )
               => ( Y2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa))))),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),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),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),product_Pair(int,int,X),Xa)) ) ) ) ).

% and_int.pelims
tff(fact_3554_minus__numeral__div__numeral,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M2,N))) ).

% minus_numeral_div_numeral
tff(fact_3555_numeral__div__minus__numeral,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M2,N))) ).

% numeral_div_minus_numeral
tff(fact_3556_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_3557_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( aa(set(nat),set(nat),insert(nat,M2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) = set_or1337092689740270186AtMost(nat,M2,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_3558_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
     => ( set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3559_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( set_or1337092689740270186AtMost(nat,M2,N) = aa(set(nat),set(nat),insert(nat,M2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3560_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),nat_set_decode(Z)))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),Z)) = aa(set(nat),set(nat),insert(nat,N),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_3561_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
     => ( ! [K2: int,L2: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,K2),L2))
           => ( ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L2),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)))))) )
               => pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,K2),L2)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_3562_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
     => ( ! [I3: int,J: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,I3),J))
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),J))
               => pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J)) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_3563_divmod__BitM__2__eq,axiom,
    ! [M2: num] : unique8689654367752047608divmod(int,bitM(M2),aa(num,num,bit0,one2)) = aa(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),M2)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_3564_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,sinh(real),aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),aa(real,real,inverse_inverse(real),X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% sinh_ln_real
tff(fact_3565_one__mod__minus__numeral,axiom,
    ! [N: num] : modulo_modulo(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N)))) ).

% one_mod_minus_numeral
tff(fact_3566_minus__one__mod__numeral,axiom,
    ! [N: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_mod_numeral
tff(fact_3567_sinh__real__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa(real,real,sinh(real),X) = aa(real,real,sinh(real),Y2) )
    <=> ( X = Y2 ) ) ).

% sinh_real_eq_iff
tff(fact_3568_sinh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,sinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sinh_0
tff(fact_3569_sinh__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,sinh(A),aa(A,A,uminus_uminus(A),X)) = aa(A,A,uminus_uminus(A),aa(A,A,sinh(A),X)) ) ).

% sinh_minus
tff(fact_3570_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sinh(real),X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% sinh_real_zero_iff
tff(fact_3571_sinh__real__less__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sinh(real),X)),aa(real,real,sinh(real),Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2)) ) ).

% sinh_real_less_iff
tff(fact_3572_sinh__real__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sinh(real),X)),aa(real,real,sinh(real),Y2)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y2)) ) ).

% sinh_real_le_iff
tff(fact_3573_sinh__real__abs,axiom,
    ! [X: real] : aa(real,real,sinh(real),aa(real,real,abs_abs(real),X)) = aa(real,real,abs_abs(real),aa(real,real,sinh(real),X)) ).

% sinh_real_abs
tff(fact_3574_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sinh(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% sinh_real_neg_iff
tff(fact_3575_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sinh(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% sinh_real_pos_iff
tff(fact_3576_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sinh(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% sinh_real_nonpos_iff
tff(fact_3577_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sinh(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% sinh_real_nonneg_iff
tff(fact_3578_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_3579_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_3580_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_mod_numeral
tff(fact_3581_minus__numeral__mod__numeral,axiom,
    ! [M2: num,N: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M2,N))) ).

% minus_numeral_mod_numeral
tff(fact_3582_numeral__mod__minus__numeral,axiom,
    ! [M2: num,N: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),M2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M2,N)))) ).

% numeral_mod_minus_numeral
tff(fact_3583_arsinh__sinh__real,axiom,
    ! [X: real] : aa(real,real,arsinh(real),aa(real,real,sinh(real),X)) = X ).

% arsinh_sinh_real
tff(fact_3584_sinh__less__cosh__real,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sinh(real),X)),aa(real,real,cosh(real),X))) ).

% sinh_less_cosh_real
tff(fact_3585_sinh__le__cosh__real,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sinh(real),X)),aa(real,real,cosh(real),X))) ).

% sinh_le_cosh_real
tff(fact_3586_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = aa(num,num,bit0,N) ).

% inc_BitM_eq
tff(fact_3587_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_3588_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa(num,num,bit1,N) ).

% BitM_inc_eq
tff(fact_3589_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(N))) ).

% eval_nat_numeral(2)
tff(fact_3590_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,cosh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = 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,cosh(A),X)),aa(A,A,cosh(A),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sinh(A),X)),aa(A,A,sinh(A),Y2))) ) ).

% cosh_add
tff(fact_3591_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,sinh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = 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,sinh(A),X)),aa(A,A,cosh(A),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cosh(A),X)),aa(A,A,sinh(A),Y2))) ) ).

% sinh_add
tff(fact_3592_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,sinh(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sinh(A),X)),aa(A,A,cosh(A),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cosh(A),X)),aa(A,A,sinh(A),Y2))) ) ).

% sinh_diff
tff(fact_3593_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : aa(A,A,cosh(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cosh(A),X)),aa(A,A,cosh(A),Y2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sinh(A),X)),aa(A,A,sinh(A),Y2))) ) ).

% cosh_diff
tff(fact_3594_BitM__plus__one,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(N)),one2) = aa(num,num,bit0,N) ).

% BitM_plus_one
tff(fact_3595_one__plus__BitM,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(N)) = aa(num,num,bit0,N) ).

% one_plus_BitM
tff(fact_3596_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,sinh(A),X)),aa(A,A,cosh(A),X)) = aa(A,A,exp(A),X) ) ).

% sinh_plus_cosh
tff(fact_3597_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,cosh(A),X)),aa(A,A,sinh(A),X)) = aa(A,A,exp(A),X) ) ).

% cosh_plus_sinh
tff(fact_3598_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tanh(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sinh(A),X)),aa(A,A,cosh(A),X)) ) ).

% tanh_def
tff(fact_3599_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bitM(N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))),one_one(A)) ) ).

% numeral_BitM
tff(fact_3600_cosh__minus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,cosh(A),X)),aa(A,A,sinh(A),X)) = aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)) ) ).

% cosh_minus_sinh
tff(fact_3601_sinh__minus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,sinh(A),X)),aa(A,A,cosh(A),X)) = aa(A,A,uminus_uminus(A),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X))) ) ).

% sinh_minus_cosh
tff(fact_3602_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,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))),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,sinh(A),X))),aa(A,A,cosh(A),X)) ) ).

% sinh_double
tff(fact_3603_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,sinh(A),X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,exp(A),X)),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_3604_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : aa(A,A,sinh(A),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sinh_field_def
tff(fact_3605_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cosh(A),X)),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),aa(A,A,sinh(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_3606_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sinh(A),X)),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),aa(A,A,cosh(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_3607_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: 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,cosh(A),X)),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,sinh(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_3608_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,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))),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cosh(A),X)),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,sinh(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cosh_double
tff(fact_3609_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ).

% Suc_0_xor_eq
tff(fact_3610_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ).

% xor_Suc_0_eq
tff(fact_3611_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list(bool)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(list(bool),int,aa(int,fun(list(bool),int),aa(fun(bool,int),fun(int,fun(list(bool),int)),groups4207007520872428315er_sum(bool,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(bool),nat,size_size(list(bool)),Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_3612_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).

% push_bit_numeral_minus_1
tff(fact_3613_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa(nat,bool,vEBT_vebt_member(X),Xa))
     => ( ! [A6: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A6,B4) )
           => ( ( ( Xa = zero_zero(nat) )
               => pp(A6) )
              & ( ( Xa != zero_zero(nat) )
               => ( ( ( Xa = one_one(nat) )
                   => pp(B4) )
                  & ( Xa = one_one(nat) ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy,Uz2)
           => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( ( Xa != Mi )
                     => ( ( Xa != Ma2 )
                       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                   => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,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_VEBT_low(Xa,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)))))) )
                                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3614_set__vebt_H__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_VEBT_set_vebt(T2) = collect(nat,vEBT_vebt_member(T2)) ).

% set_vebt'_def
tff(fact_3615_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_3616_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% push_bit_negative_int_iff
tff(fact_3617_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),X) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_3618_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_3619_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_3620_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_3621_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_3622_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_3623_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : bit_concat_bit(N,zero_zero(int),L) = aa(int,int,bit_se4730199178511100633sh_bit(int,N),L) ).

% concat_bit_of_zero_1
tff(fact_3624_case__nat__numeral,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,A),V2: num] : case_nat(A,A3,F2,aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,F2,pred_numeral(V2)) ).

% case_nat_numeral
tff(fact_3625_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) ) ).

% push_bit_Suc_numeral
tff(fact_3626_case__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,A),V2: num,N: nat] : case_nat(A,A3,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N)) ).

% case_nat_add_eq_if
tff(fact_3627_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_3628_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)) ) ).

% xor_numerals(8)
tff(fact_3629_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% xor_numerals(5)
tff(fact_3630_xor__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y2: 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,Y2))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y2)) ) ).

% xor_numerals(2)
tff(fact_3631_xor__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y2: 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,Y2))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y2)) ) ).

% xor_numerals(1)
tff(fact_3632_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),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_3633_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ).

% push_bit_of_1
tff(fact_3634_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% push_bit_of_Suc_0
tff(fact_3635_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3)))
        <=> ( ( N != zero_zero(nat) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) ) ) ) ).

% even_push_bit_iff
tff(fact_3636_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_3637_xor__nat__numerals_I1_J,axiom,
    ! [Y2: 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,Y2))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y2)) ).

% xor_nat_numerals(1)
tff(fact_3638_xor__nat__numerals_I2_J,axiom,
    ! [Y2: 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,Y2))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y2)) ).

% xor_nat_numerals(2)
tff(fact_3639_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_3640_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_3641_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_3642_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y2: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y2))) = 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),X)),aa(num,A,numeral_numeral(A),Y2)))) ) ).

% xor_numerals(4)
tff(fact_3643_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y2: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y2))) = 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),X)),aa(num,A,numeral_numeral(A),Y2)))) ) ).

% xor_numerals(6)
tff(fact_3644_Compl__eq,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = collect(A,aTP_Lamp_aa(set(A),fun(A,bool),A4)) ).

% Compl_eq
tff(fact_3645_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] : collect(A,aTP_Lamp_ab(fun(A,bool),fun(A,bool),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).

% Collect_neg_eq
tff(fact_3646_uminus__set__def,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = collect(A,aa(fun(A,bool),fun(A,bool),uminus_uminus(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A4))) ).

% uminus_set_def
tff(fact_3647_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3)) ) ).

% push_bit_minus
tff(fact_3648_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_code(2)
tff(fact_3649_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),collect(A,aTP_Lamp_ac(A,fun(A,bool),A3))),collect(A,aTP_Lamp_ac(A,fun(A,bool),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3650_flip__bit__nat__def,axiom,
    ! [M2: nat,N: nat] : bit_se8732182000553998342ip_bit(nat,M2,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M2),one_one(nat))) ).

% flip_bit_nat_def
tff(fact_3651_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ad(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_3652_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_3653_nat_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(nat,A),Nat: nat] : aa(A,B,H,case_nat(A,F1,F22,Nat)) = case_nat(B,aa(A,B,H,F1),aa(fun(nat,A),fun(nat,B),aTP_Lamp_ae(fun(A,B),fun(fun(nat,A),fun(nat,B)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_3654_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M2: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M2)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),M2)) ) ).

% push_bit_of_nat
tff(fact_3655_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M2),N)) = aa(A,A,bit_se4730199178511100633sh_bit(A,M2),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_push_bit
tff(fact_3656_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_xor_eq
tff(fact_3657_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> pp(case_nat(bool,fFalse,aTP_Lamp_af(nat,bool),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_3658_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> pp(case_nat(bool,fTrue,aTP_Lamp_ag(nat,bool),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_3659_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ah(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_3660_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_ai(nat,nat),Nat) ).

% pred_def
tff(fact_3661_less__eq__nat_Osimps_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N))
    <=> pp(case_nat(bool,fFalse,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_3662_set__diff__eq,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_aj(set(A),fun(set(A),fun(A,bool)),A4),B5)) ).

% set_diff_eq
tff(fact_3663_minus__set__def,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A4)),aTP_Lamp_a(set(A),fun(A,bool),B5))) ).

% minus_set_def
tff(fact_3664_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_ac(A,fun(A,bool),A3))),collect(A,aTP_Lamp_ac(A,fun(A,bool),B2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% subset_divisors_dvd
tff(fact_3665_diff__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_ai(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) ).

% diff_Suc
tff(fact_3666_nat__less__as__int,axiom,
    ! [X3: nat,Xa2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ) ).

% nat_less_as_int
tff(fact_3667_nat__leq__as__int,axiom,
    ! [X3: nat,Xa2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ) ).

% nat_leq_as_int
tff(fact_3668_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_code(3)
tff(fact_3669_set__vebt__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_set_vebt(T2) = collect(nat,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2)) ).

% set_vebt_def
tff(fact_3670_nat__plus__as__int,axiom,
    ! [X3: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X3),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_plus_as_int
tff(fact_3671_nat__times__as__int,axiom,
    ! [X3: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X3),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_times_as_int
tff(fact_3672_nat__minus__as__int,axiom,
    ! [X3: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X3),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_minus_as_int
tff(fact_3673_nat__div__as__int,axiom,
    ! [X3: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X3),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_div_as_int
tff(fact_3674_nat__mod__as__int,axiom,
    ! [X3: nat,Xa2: nat] : modulo_modulo(nat,X3,Xa2) = aa(int,nat,nat2,modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X3),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_mod_as_int
tff(fact_3675_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,N,A3) = if(A,aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3)),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3)),aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A)))),bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3)) ) ).

% signed_take_bit_code
tff(fact_3676_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat,A3: A] : bit_se2584673776208193580ke_bit(A,M2,aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3)) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),A3)) ) ).

% take_bit_push_bit
tff(fact_3677_diff__nat__eq__if,axiom,
    ! [Z2: int,Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) = aa(int,nat,nat2,Z) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) = if(nat,aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z2)),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z2))) ) ) ) ).

% diff_nat_eq_if
tff(fact_3678_bit__push__bit__iff__int,axiom,
    ! [M2: nat,K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,M2),K)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ).

% bit_push_bit_iff_int
tff(fact_3679_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect(nat,aTP_Lamp_ak(nat,fun(nat,bool),X)) ).

% set_decode_def
tff(fact_3680_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q5: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M2),Q5)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,Q5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_3681_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_al(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),one_one(A)) ) ) ) ) ).

% pochhammer_code
tff(fact_3682_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_3683_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( ( ( K = zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A3),K) = one_one(A) ) )
          & ( ( K != zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_am(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_3684_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList,S),X)
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_3685_push__bit__minus__one,axiom,
    ! [N: nat] : aa(int,int,bit_se4730199178511100633sh_bit(int,N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% push_bit_minus_one
tff(fact_3686_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd2),X)
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3687_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi2: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),aa(nat,nat,suc,V2),TreeList,Vc),X)
    <=> ( ( X = Mi2 )
        | ( X = Ma )
        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3688_vebt__member_Osimps_I5_J,axiom,
    ! [Mi2: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X))
    <=> ( ( X != Mi2 )
       => ( ( X != Ma )
         => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi2))
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi2))
             => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
                & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
                 => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                     => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_3689_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa)
     => ( ! [A6: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A6,B4) )
           => ( ( ( Xa = zero_zero(nat) )
               => pp(A6) )
              & ( ( Xa != zero_zero(nat) )
               => ( ( ( Xa = one_one(nat) )
                   => pp(B4) )
                  & ( Xa = one_one(nat) ) ) ) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2)
               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3690_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(X,Xa)
     => ( ! [A6: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A6,B4) )
           => ~ ( ( ( Xa = zero_zero(nat) )
                 => pp(A6) )
                & ( ( Xa != zero_zero(nat) )
                 => ( ( ( Xa = one_one(nat) )
                     => pp(B4) )
                    & ( Xa = one_one(nat) ) ) ) ) )
       => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2)
             => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3691_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa)
      <=> pp(Y2) )
     => ( ! [A6: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A6,B4) )
           => ( pp(Y2)
            <=> ~ ( ( ( Xa = zero_zero(nat) )
                   => pp(A6) )
                  & ( ( Xa != zero_zero(nat) )
                   => ( ( ( Xa = one_one(nat) )
                       => pp(B4) )
                      & ( Xa = one_one(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => pp(Y2) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2)
               => ( pp(Y2)
                <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3692_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(X,Xa)
     => ( ! [Mi: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xa = Mi )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)
             => ~ ( ( Xa = Mi )
                  | ( Xa = Ma2 )
                  | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) )
         => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd)
               => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3693_xor__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N) = N ) )
      & ( ( M2 != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N) = M2 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,N,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),M2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_3694_xor__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2))),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))))),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),M2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% xor_nat_rec
tff(fact_3695_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(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)))) ) ).

% xor_one_eq
tff(fact_3696_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(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)))) ) ).

% one_xor_eq
tff(fact_3697_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa))
     => ( ! [A6: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A6,B4) )
           => ~ ( ( ( Xa = zero_zero(nat) )
                 => pp(A6) )
                & ( ( Xa != zero_zero(nat) )
                 => ( ( ( Xa = one_one(nat) )
                     => pp(B4) )
                    & ( Xa = one_one(nat) ) ) ) ) )
       => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
             => ~ ( ( Xa != Mi )
                 => ( ( Xa != Ma2 )
                   => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                           => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                               => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,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_VEBT_low(Xa,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)))))) )
                              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3698_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa)
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( ( Xa = Mi )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)
                 => ( ( Xa = Mi )
                    | ( Xa = Ma2 )
                    | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) )
             => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd)
                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3699_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa)
      <=> pp(Y2) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y2) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)
           => pp(Y2) )
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( pp(Y2)
                <=> ~ ( ( Xa = Mi )
                      | ( Xa = Ma2 ) ) ) )
           => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)
                 => ( pp(Y2)
                  <=> ~ ( ( Xa = Mi )
                        | ( Xa = Ma2 )
                        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) )
             => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd)
                   => ( pp(Y2)
                    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3700_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list(bool),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Bs)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(bool),nat,size_size(list(bool)),Bs)))
            & pp(aa(nat,bool,nth(bool,Bs),N)) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_3701_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: bool] :
      ( ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa))
      <=> pp(Y2) )
     => ( ! [A6: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A6,B4) )
           => ( pp(Y2)
            <=> ~ ( ( ( Xa = zero_zero(nat) )
                   => pp(A6) )
                  & ( ( Xa != zero_zero(nat) )
                   => ( ( ( Xa = one_one(nat) )
                       => pp(B4) )
                      & ( Xa = one_one(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => pp(Y2) )
         => ( ( ? [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy,Uz2)
             => pp(Y2) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => pp(Y2) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( pp(Y2)
                    <=> ~ ( ( Xa != Mi )
                         => ( ( Xa != Ma2 )
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                               => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                                  & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                       => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,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_VEBT_low(Xa,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)))))) )
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3702_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ( ( ( K = zero_zero(int) )
           => ( aa(int,A,ring_1_of_int(A),K) = zero_zero(A) ) )
          & ( ( K != zero_zero(int) )
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
               => ( aa(int,A,ring_1_of_int(A),K) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K))) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
               => ( aa(int,A,ring_1_of_int(A),K) = if(A,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)),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))))),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(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)))))),one_one(A))) ) ) ) ) ) ) ).

% of_int_code_if
tff(fact_3703_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => topological_monoseq(real,aTP_Lamp_an(real,fun(nat,real),X)) ) ).

% monoseq_arctan_series
tff(fact_3704_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ao(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))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_3705_ln__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,real,ln_ln(real),X) = suminf(real,aTP_Lamp_ap(real,fun(nat,real),X)) ) ) ) ).

% ln_series
tff(fact_3706_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_3707_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int)))
    <=> ~ ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% xor_negative_int_iff
tff(fact_3708_Ints__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ring_1(B) )
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
             => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X2)),ring_1_Ints(B))) )
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),ring_1_Ints(B))) ) ) ).

% Ints_prod
tff(fact_3709_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,nat),A4: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F2),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_aq(fun(B,nat),fun(B,A),F2)),A4) ) ).

% of_nat_prod
tff(fact_3710_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F2: fun(B,int),A4: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),F2),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ar(fun(B,int),fun(B,A),F2)),A4) ) ).

% of_int_prod
tff(fact_3711_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_as(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_3712_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M2: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = 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,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_3713_XOR__lower,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y2))) ) ) ).

% XOR_lower
tff(fact_3714_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_at(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_3715_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se8732182000553998342ip_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,bit_se4730199178511100633sh_bit(int,N),one_one(int))) ).

% flip_bit_int_def
tff(fact_3716_xor__nat__def,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% xor_nat_def
tff(fact_3717_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,N,M2)) = 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_au(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M2)),set_or1337092689740270186AtMost(nat,N,M2)) ) ).

% prod.atLeastAtMost_rev
tff(fact_3718_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_3719_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3720_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3721_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( 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,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_at(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3722_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ai(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).

% fact_prod
tff(fact_3723_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_av(fun(nat,A),fun(nat,fun(A,A)),F2),A3,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_3724_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),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,M2,N))),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),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3725_fact__eq__fact__times,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( semiring_char_0_fact(nat,M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ai(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M2))) ) ) ).

% fact_eq_fact_times
tff(fact_3726_monoseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => topological_monoseq(real,aa(real,fun(nat,real),power_power(real),X)) ) ) ).

% monoseq_realpow
tff(fact_3727_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aw(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_3728_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ax(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_3729_fact__div__fact,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M2)),semiring_char_0_fact(nat,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ai(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M2)) ) ) ).

% fact_div_fact
tff(fact_3730_XOR__upper,axiom,
    ! [X: int,N: nat,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ) ).

% XOR_upper
tff(fact_3731_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: 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))),M2),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(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ay(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.in_pairs
tff(fact_3732_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ax(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3733_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_az(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_3734_xor__int__rec,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K))),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),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_3735_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_ba(nat,real)) ).

% pi_series
tff(fact_3736_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,arctan,X) = suminf(real,aTP_Lamp_bb(real,fun(nat,real),X)) ) ) ).

% arctan_series
tff(fact_3737_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => ( suminf(A,aa(A,fun(nat,A),power_power(A),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_3738_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_3739_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_bc(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_3740_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_bd(B,A)),A4) = one_one(A) ) ).

% prod.neutral_const
tff(fact_3741_int__prod,axiom,
    ! [B: $tType,F2: fun(B,nat),A4: set(B)] : aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F2),A4)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),aTP_Lamp_be(fun(B,nat),fun(B,int),F2)),A4) ).

% int_prod
tff(fact_3742_prod__int__eq,axiom,
    ! [I: nat,J2: 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,J2)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_bf(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J2))) ).

% prod_int_eq
tff(fact_3743_prod__int__plus__eq,axiom,
    ! [I: nat,J2: 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),J2))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_bf(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),J2)))) ).

% prod_int_plus_eq
tff(fact_3744_prod_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),G: fun(B,A)] :
          ( ! [X2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
             => ( aa(B,A,G,X2) = one_one(A) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = one_one(A) ) ) ) ).

% prod.neutral
tff(fact_3745_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) )
         => ~ ! [A6: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),A4))
               => ( aa(B,A,G,A6) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_3746_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_bg(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_3747_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A4: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),aa(B,A,G,I3))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(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_mono
tff(fact_3748_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A4: set(B),F2: fun(B,A)] :
          ( ! [X2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4))) ) ) ).

% prod_nonneg
tff(fact_3749_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A4: set(B),F2: fun(B,A)] :
          ( ! [X2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F2,X2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4))) ) ) ).

% prod_pos
tff(fact_3750_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A4: set(B),F2: fun(B,A)] :
          ( ! [X2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F2,X2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4))) ) ) ).

% prod_ge_1
tff(fact_3751_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A4: set(B),F2: fun(B,A)] :
          ( ! [X2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X2)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X2)),one_one(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4)),one_one(A))) ) ) ).

% prod_le_1
tff(fact_3752_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,root(N),real_V7770717601297561774m_norm(complex,C2)))),cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),arg(C2)),aa(nat,real,semiring_1_of_nat(real),N))))),collect(complex,aTP_Lamp_bh(nat,fun(complex,bool),N)),collect(complex,aa(nat,fun(complex,bool),aTP_Lamp_bi(complex,fun(nat,fun(complex,bool)),C2),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_3753_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => summable(real,aTP_Lamp_bb(real,fun(nat,real),X)) ) ).

% summable_arctan_series
tff(fact_3754_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),L) ) )
      & ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),K) ) )
          & ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( K = zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = L ) )
              & ( ( K != zero_zero(int) )
               => ( ( ( L = zero_zero(int) )
                   => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = K ) )
                  & ( ( L != zero_zero(int) )
                   => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,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_3755_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y2: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y2 )
     => ( ( ( X = zero_zero(nat) )
         => ( Y2 != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y2 != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
                     => ( Y2 = 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,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_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) )
                    & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
                     => ( Y2 = 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,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_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_3756_intind,axiom,
    ! [A: $tType,I: nat,N: nat,P: fun(A,bool),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
     => ( pp(aa(A,bool,P,X))
       => pp(aa(A,bool,P,aa(nat,A,nth(A,replicate(A,N,X)),I))) ) ) ).

% intind
tff(fact_3757_replicate__eq__replicate,axiom,
    ! [A: $tType,M2: nat,X: A,N: nat,Y2: A] :
      ( ( replicate(A,M2,X) = replicate(A,N,Y2) )
    <=> ( ( M2 = N )
        & ( ( M2 != zero_zero(nat) )
         => ( X = Y2 ) ) ) ) ).

% replicate_eq_replicate
tff(fact_3758_length__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),nat,size_size(list(A)),replicate(A,N,X)) = N ).

% length_replicate
tff(fact_3759_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bj(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_3760_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_bk(nat,A)) ) ).

% summable_zero
tff(fact_3761_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_3762_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_3763_in__set__replicate,axiom,
    ! [A: $tType,X: A,N: nat,Y2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),replicate(A,N,Y2))))
    <=> ( ( X = Y2 )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_3764_Bex__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,P: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),replicate(A,N,A3))))
          & pp(aa(A,bool,P,X4)) )
    <=> ( pp(aa(A,bool,P,A3))
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_3765_Ball__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),replicate(A,N,A3))))
         => pp(aa(A,bool,P,X4)) )
    <=> ( pp(aa(A,bool,P,A3))
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_3766_nth__replicate,axiom,
    ! [A: $tType,I: nat,N: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
     => ( aa(nat,A,nth(A,replicate(A,N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_3767_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_bl(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_3768_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_bm(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_3769_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_3770_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_3771_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_left
tff(fact_3772_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_right
tff(fact_3773_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_3774_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_3775_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% not_nonnegative_int_iff
tff(fact_3776_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% not_negative_int_iff
tff(fact_3777_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),inc(N)) ) ).

% minus_not_numeral_eq
tff(fact_3778_set__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_3779_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_3780_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,aa(A,fun(nat,A),power_power(A),C2))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))) ) ) ).

% summable_geometric_iff
tff(fact_3781_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_3782_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_minus_iff
tff(fact_3783_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => summable(A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),F2)) ) ) ).

% summable_minus
tff(fact_3784_summable__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_diff
tff(fact_3785_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_bp(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_3786_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_bm(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_3787_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bq(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_Suc_iff
tff(fact_3788_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X)))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_bs(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_3789_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_3790_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_3791_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
     => ( ! [Y3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),aa(list(A),set(A),set2(A),Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate(A,N,X) ) ) ) ).

% replicate_eqI
tff(fact_3792_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
         => ( X2 = X ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_3793_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_bl(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_3794_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_3795_suminf__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_diff
tff(fact_3796_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_bm(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_3797_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),F2)) = aa(A,A,uminus_uminus(A),suminf(A,F2)) ) ) ) ).

% suminf_minus
tff(fact_3798_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N5: nat] : aa(nat,A,F2,N5) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3799_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),suminf(A,F2))) ) ) ) ).

% suminf_nonneg
tff(fact_3800_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F2))) ) ) ) ).

% suminf_pos
tff(fact_3801_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_3802_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_3803_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_3804_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_3805_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bu(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_3806_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_bv(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_bw(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% powser_split_head(3)
tff(fact_3807_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_bx(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_3808_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_3809_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_3810_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A3: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2))
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N2)) )
         => ( 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_3811_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_se2584673776208193580ke_bit(A,N,aa(A,A,bit_ri4277139882892585799ns_not(A),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),bit_se2239418461657761734s_mask(A,N)),bit_se2584673776208193580ke_bit(A,N,A3)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_3812_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N)) ) ).

% minus_numeral_inc_eq
tff(fact_3813_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F2)))
            <=> ? [I2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,I2))) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3814_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I: nat] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,I)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F2))) ) ) ) ) ).

% suminf_pos2
tff(fact_3815_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_3816_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bv(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X)))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_bv(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_3817_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_3818_and__not__numerals_I4_J,axiom,
    ! [M2: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,M2)) ).

% and_not_numerals(4)
tff(fact_3819_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))) ) ).

% not_numeral_Bit0_eq
tff(fact_3820_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => summable(A,aa(A,fun(nat,A),power_power(A),C2)) ) ) ).

% summable_geometric
tff(fact_3821_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => summable(A,aa(A,fun(nat,A),power_power(A),X)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3822_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bq(fun(nat,A),fun(nat,A),F2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3823_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : summable(A,aTP_Lamp_by(A,fun(nat,A),X)) ) ).

% summable_exp
tff(fact_3824_set__replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,N),X)) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_3825_set__replicate__conv__if,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = bot_bot(set(A)) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ).

% set_replicate_conv_if
tff(fact_3826_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K)),N))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))),N)) ) ).

% bit_minus_int_iff
tff(fact_3827_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))) ) ).

% not_numeral_BitM_eq
tff(fact_3828_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( bit_se2584673776208193580ke_bit(A,M2,aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_3829_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,N,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,N),one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_3830_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se2638667681897837118et_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,bit_se4730199178511100633sh_bit(int,N),one_one(int)))) ).

% unset_bit_int_def
tff(fact_3831_and__not__numerals_I7_J,axiom,
    ! [M2: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,M2)) ).

% and_not_numerals(7)
tff(fact_3832_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_3833_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_bv(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_bv(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_bw(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3834_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_bv(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_bw(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_bv(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3835_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,I3)),one_one(real)))
     => ( ! [I3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,I3)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Z))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z),one_one(real)))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_bz(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3836_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),N))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) )
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ) ).

% bit_not_iff_eq
tff(fact_3837_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% minus_exp_eq_not_mask
tff(fact_3838_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N6: nat,F2: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),one_one(real)))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N2))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,aa(nat,nat,suc,N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))))) )
           => summable(A,F2) ) ) ) ).

% summable_ratio_test
tff(fact_3839_and__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% and_not_numerals(8)
tff(fact_3840_not__int__rec,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(int,bool,aa(int,fun(int,bool),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_3841_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_3842_sin__paired,axiom,
    ! [X: real] : sums(real,aTP_Lamp_ca(real,fun(nat,real),X),aa(real,real,sin(real),X)) ).

% sin_paired
tff(fact_3843_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa(nat,bool,vEBT_vebt_member(X),Xa))
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [A6: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A6,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A6,B4)),Xa))
               => ( ( ( Xa = zero_zero(nat) )
                   => pp(A6) )
                  & ( ( Xa != zero_zero(nat) )
                   => ( ( ( Xa = one_one(nat) )
                       => pp(B4) )
                      & ( Xa = one_one(nat) ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xa)) )
           => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy,Uz2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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)),V4),zero_zero(nat),Uy,Uz2)),Xa)) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa)) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa))
                       => ( ( Xa != Mi )
                         => ( ( Xa != Ma2 )
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                               => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                                  & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                       => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,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_VEBT_low(Xa,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)))))) )
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3844_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: bool] :
      ( ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa))
      <=> pp(Y2) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [A6: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A6,B4) )
             => ( ( pp(Y2)
                <=> ( ( ( Xa = zero_zero(nat) )
                     => pp(A6) )
                    & ( ( Xa != zero_zero(nat) )
                     => ( ( ( Xa = one_one(nat) )
                         => pp(B4) )
                        & ( Xa = one_one(nat) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A6,B4)),Xa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ pp(Y2)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy,Uz2) )
                 => ( ~ pp(Y2)
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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)),V4),zero_zero(nat),Uy,Uz2)),Xa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ pp(Y2)
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( pp(Y2)
                        <=> ( ( Xa != Mi )
                           => ( ( Xa != Ma2 )
                             => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                                & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                                 => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                                    & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                                     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                         => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,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_VEBT_low(Xa,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)))))) )
                                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3845_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [N: nat,M2: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( ( ( X = one_one(A) )
               => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N)) = 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),N),one_one(nat))),M2)) ) )
              & ( ( X != one_one(A) )
               => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N)) = 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),X),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ) ) ).

% sum_gp
tff(fact_3846_Ints__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
             => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X2)),ring_1_Ints(B))) )
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),ring_1_Ints(B))) ) ) ).

% Ints_sum
tff(fact_3847_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cb(B,A)),A4) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_3848_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F2: fun(B,nat),A4: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F2),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cc(fun(B,nat),fun(B,A),F2)),A4) ) ).

% of_nat_sum
tff(fact_3849_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F2: fun(B,int),A4: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),F2),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cd(fun(B,int),fun(B,A),F2)),A4) ) ).

% of_int_sum
tff(fact_3850_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_bk(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_3851_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_3852_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F2: fun(A,B),A4: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_ce(fun(A,B),fun(A,B),F2)),A4))) ) ).

% sum_abs_ge_zero
tff(fact_3853_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A),X: A] :
          ( sums(A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),A3),X)
        <=> ( aa(nat,A,A3,zero_zero(nat)) = X ) ) ) ).

% powser_sums_zero_iff
tff(fact_3854_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M2: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_3855_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_bm(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_3856_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bj(nat,fun(fun(nat,A),fun(nat,A)),I),F2),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_3857_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A4: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) != zero_zero(A) )
         => ~ ! [A6: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),A4))
               => ( aa(B,A,G,A6) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_3858_sum_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),G: fun(B,A)] :
          ( ! [X2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
             => ( aa(B,A,G,X2) = zero_zero(A) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = zero_zero(A) ) ) ) ).

% sum.neutral
tff(fact_3859_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] :
          ( ! [N2: nat] : aa(nat,A,F2,N2) = zero_zero(A)
         => sums(A,F2,zero_zero(A)) ) ) ).

% sums_0
tff(fact_3860_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(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_bo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% sums_diff
tff(fact_3861_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A3: A] :
          ( sums(A,F2,A3)
         => sums(A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),F2),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% sums_minus
tff(fact_3862_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cf(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A4) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)) ) ).

% sum_subtractf
tff(fact_3863_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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),R) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_cg(fun(B,A),fun(A,fun(B,A)),F2),R)),A4) ) ).

% sum_divide_distrib
tff(fact_3864_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ch(fun(B,A),fun(B,A),F2)),A4) = aa(A,A,uminus_uminus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)) ) ).

% sum_negf
tff(fact_3865_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A4: set(B),F2: fun(B,A)] :
          ( ! [X2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4))) ) ) ).

% sum_nonneg
tff(fact_3866_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A4: set(B),F2: fun(B,A)] :
          ( ! [X2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X2)),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),zero_zero(A))) ) ) ).

% sum_nonpos
tff(fact_3867_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4))
         => ( ! [X2: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,X2)),A4))
               => ( aa(nat,A,F2,aa(nat,nat,suc,X2)) = aa(nat,A,G,aa(nat,nat,suc,X2)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A4) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A4) ) ) ) ) ).

% sum_cong_Suc
tff(fact_3868_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_ci(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_3869_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_cj(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_3870_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_3871_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_bl(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_3872_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_bq(fun(nat,A),fun(nat,A),F2),S)
           => sums(A,F2,S) ) ) ) ).

% sums_Suc_imp
tff(fact_3873_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_cl(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_3874_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,aTP_Lamp_bq(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_3875_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [N: nat,F2: fun(nat,A),S: A] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
             => ( aa(nat,A,F2,I3) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cm(nat,fun(fun(nat,A),fun(nat,A)),N),F2),S)
          <=> sums(A,F2,S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3876_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,N,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_cn(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M2)),set_or1337092689740270186AtMost(nat,N,M2)) ) ).

% sum.atLeastAtMost_rev
tff(fact_3877_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M2: nat,Z: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_co(nat,fun(A,fun(nat,A)),M2),Z),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M2)) ) ).

% powser_sums_if
tff(fact_3878_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A)] : sums(A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),A3),aa(nat,A,A3,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3879_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_3880_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_3881_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_3882_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_3883_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_3884_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cp(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,M2)) ) ) ) ).

% sum_Suc_diff
tff(fact_3885_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_cq(fun(nat,A),fun(nat,fun(A,A)),F2),A3,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3886_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_3887_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
          ( ! [I3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I3))) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I5) = one_one(B) )
           => ( ! [I3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),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,I3)),B2))),Delta)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_cr(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I5)),B2))),Delta)) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3888_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F2: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cs(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,M2)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cs(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M2,N)) = zero_zero(A) ) ) ) ) ).

% sum_natinterval_diff
tff(fact_3889_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ct(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,N)),aa(nat,A,F2,M2)) ) ) ) ).

% sum_telescope''
tff(fact_3890_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),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_3891_power__half__series,axiom,
    sums(real,aTP_Lamp_cu(nat,real),one_one(real)) ).

% power_half_series
tff(fact_3892_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: 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))),N)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(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)))),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N))) ) ).

% mask_eq_sum_exp
tff(fact_3893_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M2: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( 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)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3894_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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))),M2),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(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cw(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.in_pairs
tff(fact_3895_sums__if_H,axiom,
    ! [G: fun(nat,real),X: real] :
      ( sums(real,G,X)
     => sums(real,aTP_Lamp_cx(fun(nat,real),fun(nat,real),G),X) ) ).

% sums_if'
tff(fact_3896_sums__if,axiom,
    ! [G: fun(nat,real),X: real,F2: fun(nat,real),Y2: real] :
      ( sums(real,G,X)
     => ( sums(real,F2,Y2)
       => sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_cy(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y2)) ) ) ).

% sums_if
tff(fact_3897_mask__eq__sum__exp__nat,axiom,
    ! [N: 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))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(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)))),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N))) ).

% mask_eq_sum_exp_nat
tff(fact_3898_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ai(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% gauss_sum_nat
tff(fact_3899_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cz(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_3900_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,D2: A,N: 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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_da(A,fun(A,fun(nat,A)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),N)),D2))) ) ).

% double_arith_series
tff(fact_3901_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_3902_arith__series__nat,axiom,
    ! [A3: nat,D2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_db(nat,fun(nat,fun(nat,nat)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D2)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% arith_series_nat
tff(fact_3903_Sum__Icc__nat,axiom,
    ! [M2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ai(nat,nat)),set_or1337092689740270186AtMost(nat,M2,N)) = 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),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Icc_nat
tff(fact_3904_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum
tff(fact_3905_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,D2: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dc(A,fun(A,fun(nat,A)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),N)),D2)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% arith_series
tff(fact_3906_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_3907_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M2: nat,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),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),X),aa(nat,nat,suc,N))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_offset
tff(fact_3908_cos__paired,axiom,
    ! [X: real] : sums(real,aTP_Lamp_dd(real,fun(nat,real),X),aa(real,real,cos(real),X)) ).

% cos_paired
tff(fact_3909_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_3910_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_de(A,fun(nat,A),R)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M2)) = 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,M2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,R),aa(nat,nat,suc,M2))) ) ).

% gchoose_row_sum_weighted
tff(fact_3911_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa))
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [A6: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A6,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A6,B4)),Xa))
               => ~ ( ( ( Xa = zero_zero(nat) )
                     => pp(A6) )
                    & ( ( Xa != zero_zero(nat) )
                     => ( ( ( Xa = one_one(nat) )
                         => pp(B4) )
                        & ( Xa = one_one(nat) ) ) ) ) ) )
         => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa))
                 => ~ ( ( Xa != Mi )
                     => ( ( Xa != Ma2 )
                       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),Mi))
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa))
                               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                   => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,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_VEBT_low(Xa,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)))))) )
                                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,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(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3912_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real)))
         => sums(A,aTP_Lamp_df(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_3913_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa)
      <=> pp(Y2) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [A6: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A6,B4) )
             => ( ( pp(Y2)
                <=> ( ( ( Xa = zero_zero(nat) )
                     => pp(A6) )
                    & ( ( Xa != zero_zero(nat) )
                     => ( ( ( Xa = one_one(nat) )
                         => pp(B4) )
                        & ( Xa = one_one(nat) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A6,B4)),Xa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ pp(Y2)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xa)) ) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2) )
                 => ( ( pp(Y2)
                    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2)),Xa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3914_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(X,Xa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [A6: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A6,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A6,B4)),Xa))
               => ~ ( ( ( Xa = zero_zero(nat) )
                     => pp(A6) )
                    & ( ( Xa != zero_zero(nat) )
                     => ( ( ( Xa = one_one(nat) )
                         => pp(B4) )
                        & ( Xa = one_one(nat) ) ) ) ) ) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2)),Xa))
                 => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3915_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [A6: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A6,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A6,B4)),Xa))
               => ( ( ( Xa = zero_zero(nat) )
                   => pp(A6) )
                  & ( ( Xa != zero_zero(nat) )
                   => ( ( ( Xa = one_one(nat) )
                       => pp(B4) )
                      & ( Xa = one_one(nat) ) ) ) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xa)) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S2)),Xa))
                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3916_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(Uu2,Uv2)),Xa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)),Xa)) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),Xa))
                   => ( ( Xa = Mi )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2) )
                   => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),Xa))
                     => ( ( Xa = Mi )
                        | ( Xa = Ma2 )
                        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) )
               => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd)),Xa))
                       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3917_int__sum,axiom,
    ! [B: $tType,F2: fun(B,nat),A4: set(B)] : aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F2),A4)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),aTP_Lamp_be(fun(B,nat),fun(B,int),F2)),A4) ).

% int_sum
tff(fact_3918_sum__subtractf__nat,axiom,
    ! [A: $tType,A4: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,G,X2)),aa(A,nat,F2,X2))) )
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_dg(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,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_3919_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A4: set(A),N: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4) = aa(nat,nat,suc,N) )
     => ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X2))) ) ) ).

% sum_SucD
tff(fact_3920_sum__diff1__nat,axiom,
    ! [A: $tType,A3: A,A4: set(A),F2: fun(A,nat)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(A,nat,F2,A3)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4) ) ) ) ).

% sum_diff1_nat
tff(fact_3921_sum__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_dh(complex,complex)),collect(complex,aa(complex,fun(complex,bool),aTP_Lamp_di(nat,fun(complex,fun(complex,bool)),N),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_3922_norm__prod__diff,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [I5: set(I6),Z: fun(I6,A),W: fun(I6,A)] :
          ( ! [I3: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I3),I5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(I6,A,Z,I3))),one_one(real))) )
         => ( ! [I3: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I3),I5))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(I6,A,W,I3))),one_one(real))) )
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7121269368397514597t_prod(I6,A),Z),I5)),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7121269368397514597t_prod(I6,A),W),I5)))),aa(set(I6),real,aa(fun(I6,real),fun(set(I6),real),groups7311177749621191930dd_sum(I6,real),aa(fun(I6,A),fun(I6,real),aTP_Lamp_dj(fun(I6,A),fun(fun(I6,A),fun(I6,real)),Z),W)),I5))) ) ) ) ).

% norm_prod_diff
tff(fact_3923_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_dh(complex,complex)),collect(complex,aTP_Lamp_bh(nat,fun(complex,bool),N))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_3924_Sum__Icc__int,axiom,
    ! [M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M2),N))
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_bf(int,int)),set_or1337092689740270186AtMost(int,M2,N)) = 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),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M2),aa(int,int,aa(int,fun(int,int),minus_minus(int),M2),one_one(int))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ) ) ).

% Sum_Icc_int
tff(fact_3925_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(X,Xa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),Xa))
               => ~ ( ( Xa = Mi )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),Xa))
                 => ~ ( ( Xa = Mi )
                      | ( Xa = Ma2 )
                      | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) )
           => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd)),Xa))
                   => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3926_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa)
      <=> pp(Y2) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y2)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(Uu2,Uv2)),Xa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy) )
               => ( ~ pp(Y2)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)),Xa)) ) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( pp(Y2)
                    <=> ( ( Xa = Mi )
                        | ( Xa = Ma2 ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),Xa)) ) )
             => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2) )
                   => ( ( pp(Y2)
                      <=> ( ( Xa = Mi )
                          | ( Xa = Ma2 )
                          | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),Xa)) ) )
               => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd) )
                     => ( ( pp(Y2)
                        <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),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),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd)),Xa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3927_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => ? [T3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),T3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T3),zero_zero(real)))
            & ( aa(real,real,cos(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dk(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,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),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3928_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ? [T3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T3),X))
            & ( aa(real,real,cos(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dk(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,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),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_3929_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ? [T3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T3),X))
            & ( aa(real,real,sin(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dl(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,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),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_3930_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ? [T3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T3),X))
          & ( aa(real,real,sin(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dl(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,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),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_3931_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_ord_lessThan(A,K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),K)) ) ) ).

% lessThan_iff
tff(fact_3932_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3933_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,K),bot_bot(set(A)))),set_ord_lessThan(A,K)) = aa(set(A),set(A),insert(A,K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_3934_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,N))),aa(nat,A,G,N)) ) ).

% sum.lessThan_Suc
tff(fact_3935_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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,N))),aa(nat,A,G,N)) ) ).

% prod.lessThan_Suc
tff(fact_3936_sumr__cos__zero__one,axiom,
    ! [N: nat] : aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dm(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_3937_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = collect(A,aTP_Lamp_dn(A,fun(A,bool),U)) ) ).

% lessThan_def
tff(fact_3938_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M2: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),set_ord_lessThan(A,M2)),set_ord_lessThan(A,N)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),N)) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3939_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,K),set_ord_lessThan(nat,K)) ).

% lessThan_Suc
tff(fact_3940_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan(nat,N) = bot_bot(set(nat)) )
    <=> ( N = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_3941_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(nat,fun(nat,A)),G),N)),set_ord_lessThan(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,N)) ) ).

% sum.nat_diff_reindex
tff(fact_3942_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dp(fun(nat,A),fun(nat,fun(nat,A)),G),N)),set_ord_lessThan(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,N)) ) ).

% prod.nat_diff_reindex
tff(fact_3943_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
          ( ! [X2: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q,X2)),aa(A,nat,P,X2)))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),set_ord_lessThan(A,N))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),set_ord_lessThan(A,N))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_dq(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,N)) ) ) ) ).

% sum_diff_distrib
tff(fact_3944_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3945_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),N: nat,R: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),R)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dr(fun(nat,A),fun(A,fun(nat,A)),F2),R)),set_ord_lessThan(nat,N)) ) ).

% sumr_diff_mult_const2
tff(fact_3946_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ds(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M2)) ) ).

% sum_lessThan_telescope'
tff(fact_3947_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cp(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,M2)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3948_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N2))),X))
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3949_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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_at(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3950_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3951_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),N: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_dt(fun(nat,A),fun(nat,fun(nat,A)),F2),N),aa(A,A,aa(A,fun(A,A),minus_minus(A),S),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N))))
        <=> sums(A,F2,S) ) ) ).

% sums_iff_shift'
tff(fact_3952_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A,N: nat] :
          ( sums(A,F2,S)
         => sums(A,aa(nat,fun(nat,A),aTP_Lamp_dt(fun(nat,A),fun(nat,fun(nat,A)),F2),N),aa(A,A,aa(A,fun(A,A),minus_minus(A),S),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N)))) ) ) ).

% sums_split_initial_segment
tff(fact_3953_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_at(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3954_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_3955_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N))) ) ).

% power_diff_1_eq
tff(fact_3956_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: 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),X),N)) = 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)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N))) ) ).

% one_diff_power_eq
tff(fact_3957_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: nat] :
          ( ( X != one_one(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3958_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_dt(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_3959_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),N: nat] :
          ( summable(A,F2)
         => ( ! [M3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,M3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N))),suminf(A,F2))) ) ) ) ).

% sum_less_suminf
tff(fact_3960_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N)) = aa(nat,A,semiring_1_of_nat(A),N) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_strict
tff(fact_3961_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_du(A,fun(A,fun(nat,fun(nat,A))),Z),H),M2)),set_ord_lessThan(nat,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dv(A,fun(A,fun(nat,fun(nat,A))),Z),H),M2)),set_ord_lessThan(nat,M2)) ) ).

% lemma_termdiff1
tff(fact_3962_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dw(A,fun(nat,fun(A,fun(nat,A))),X),N),Y2)),set_ord_lessThan(nat,N))) ) ).

% power_diff_sumr2
tff(fact_3963_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y2),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dx(A,fun(nat,fun(A,fun(nat,A))),X),N),Y2)),set_ord_lessThan(nat,aa(nat,nat,suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_3964_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,F2: fun(nat,A),K6: A,K: nat] :
          ( ! [P4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P4),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,P4)),K6)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),K6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),K6))) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_3965_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),N: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M3))) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),I))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,I)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N))),suminf(A,F2))) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3966_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: 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),X),N)) = 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)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dy(A,fun(nat,fun(nat,A)),X),N)),set_ord_lessThan(nat,N))) ) ).

% one_diff_power_eq'
tff(fact_3967_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: real,N: nat,Diff: fun(nat,fun(A,real))] :
          ( ( X = zero_zero(real) )
         => ( ( N != zero_zero(nat) )
           => ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_dz(real,fun(fun(nat,fun(A,real)),fun(nat,real)),X),Diff)),set_ord_lessThan(nat,N)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_3968_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J2: fun(nat,real),N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ? [B7: real] : aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_ea(real,fun(fun(nat,real),fun(nat,real)),H),J2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),B7),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N)),semiring_char_0_fact(real,N)))) ) ).

% Maclaurin_lemma
tff(fact_3969_sum__split__even__odd,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real),N: nat] : aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_eb(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G)),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))),N))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ec(fun(nat,real),fun(nat,real),F2)),set_ord_lessThan(nat,N))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ed(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,N))) ).

% sum_split_even_odd
tff(fact_3970_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),X)))
      & ( aa(real,real,exp(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ee(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_3971_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,sin(real),X)),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dl(real,fun(nat,real),X)),set_ord_lessThan(nat,N))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_3972_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D5)))),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)))),D5)),one_one(nat)))))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),F2),set_ord_lessThan(nat,K))),suminf(real,F2))) ) ) ).

% sum_pos_lt_pair
tff(fact_3973_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero(real) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ? [T3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T3)))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,exp(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ee(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3974_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,N: 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)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_eg(A,fun(A,fun(nat,fun(nat,A))),H),Z),N)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_3975_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] : aa(real,real,sin(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dl(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,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),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ).

% Maclaurin_sin_expansion
tff(fact_3976_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),X)))
      & ( aa(real,real,sin(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dl(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,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),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_3977_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),X)))
      & ( aa(real,real,cos(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dk(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,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),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_3978_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_eh(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_ei(fun(nat,A),fun(A,fun(nat,A)),C2),X),suminf(A,aa(A,fun(nat,A),aTP_Lamp_eh(fun(nat,A),fun(A,fun(nat,A)),C2),X))) ) ) ).

% diffs_equiv
tff(fact_3979_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => bij_betw(nat,complex,aTP_Lamp_ej(nat,fun(nat,complex),N),set_ord_lessThan(nat,N),collect(complex,aTP_Lamp_bh(nat,fun(complex,bool),N))) ) ).

% bij_betw_roots_unity
tff(fact_3980_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_de(A,fun(nat,A),A3)),set_ord_atMost(nat,M2)) = 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),M2)),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),M2),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_3981_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ek(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_3982_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atMost_Suc
tff(fact_3983_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atMost_Suc
tff(fact_3984_atMost__0,axiom,
    set_ord_atMost(nat,zero_zero(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_3985_diffs__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(nat,real),X3: nat] : aa(nat,A,diffs(A,aTP_Lamp_el(fun(nat,real),fun(nat,A),F2)),X3) = real_Vector_of_real(A,aa(nat,real,diffs(real,F2),X3)) ) ).

% diffs_of_real
tff(fact_3986_atMost__atLeast0,axiom,
    ! [N: nat] : set_ord_atMost(nat,N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).

% atMost_atLeast0
tff(fact_3987_lessThan__Suc__atMost,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = set_ord_atMost(nat,K) ).

% lessThan_Suc_atMost
tff(fact_3988_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X3: nat] : aa(nat,A,diffs(A,aTP_Lamp_em(fun(nat,A),fun(nat,A),C2)),X3) = aa(A,A,uminus_uminus(A),aa(nat,A,diffs(A,C2),X3)) ) ).

% diffs_minus
tff(fact_3989_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),set_ord_atMost(nat,K)) ).

% atMost_Suc
tff(fact_3990_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,A3)),set_ord_lessThan(A,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3991_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: nat] : aa(nat,A,diffs(A,aTP_Lamp_en(nat,A)),X3) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X3)) ) ).

% exp_fdiffs
tff(fact_3992_sum__choose__upper,axiom,
    ! [M2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_eo(nat,fun(nat,nat),M2)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M2)) ).

% sum_choose_upper
tff(fact_3993_diffs__sin__coeff,axiom,
    diffs(real,sin_coeff) = cos_coeff ).

% diffs_sin_coeff
tff(fact_3994_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X3: nat] : aa(nat,A,diffs(A,C2),X3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X3))),aa(nat,A,C2,aa(nat,nat,suc,X3))) ) ).

% diffs_def
tff(fact_3995_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).

% sum.atMost_Suc_shift
tff(fact_3996_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),I: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ds(fun(nat,A),fun(nat,A),F2)),set_ord_atMost(nat,I)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_3997_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,D2: fun(nat,A)] :
          ( ! [X4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),set_ord_atMost(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),D2),X4)),set_ord_atMost(nat,N))
        <=> ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
             => ( aa(nat,A,C2,I2) = aa(nat,A,D2,I2) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_3998_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: fun(nat,A),B5: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,A3,N2)))
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A3),set_ord_atMost(nat,N2))),B5))
           => summable(A,A3) ) ) ) ).

% bounded_imp_summable
tff(fact_3999_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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_at(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_4000_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eq(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_ord_atMost(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_es(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_ord_lessThan(nat,N)) ) ).

% sum.nested_swap'
tff(fact_4001_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_et(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_ord_atMost(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ev(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_ord_lessThan(nat,N)) ) ).

% prod.nested_swap'
tff(fact_4002_sum__choose__lower,axiom,
    ! [R: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ew(nat,fun(nat,nat),R)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R),N))),N) ).

% sum_choose_lower
tff(fact_4003_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ex(nat,fun(nat,nat),N)),set_ord_atMost(nat,M2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_4004_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ex(nat,fun(nat,nat),N)),set_ord_atMost(nat,M2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),one_one(nat))),M2) ).

% choose_rising_sum(2)
tff(fact_4005_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [X2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),C2),X2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),C2),X)) ) ) ).

% termdiff_converges_all
tff(fact_4006_diffs__cos__coeff,axiom,
    ! [X3: nat] : aa(nat,real,diffs(real,cos_coeff),X3) = aa(real,real,uminus_uminus(real),aa(nat,real,sin_coeff,X3)) ).

% diffs_cos_coeff
tff(fact_4007_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),N: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),set_ord_atMost(nat,N)) = zero_zero(A)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_4008_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( ! [X4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),set_ord_atMost(nat,N)) = zero_zero(A)
        <=> ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
             => ( aa(nat,A,C2,I2) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_4009_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.atMost_shift
tff(fact_4010_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_atMost(nat,M2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)))) ) ).

% sum_up_index_split
tff(fact_4011_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_at(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% prod.atMost_shift
tff(fact_4012_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atMost(nat,N)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_4013_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fb(A,fun(nat,A),A3)),set_ord_atMost(nat,N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),N) ) ).

% gbinomial_parallel_sum
tff(fact_4014_sum__choose__diagonal,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fc(nat,fun(nat,fun(nat,nat)),M2),N)),set_ord_atMost(nat,M2)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M2) ) ) ).

% sum_choose_diagonal
tff(fact_4015_vandermonde,axiom,
    ! [M2: nat,N: nat,R: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fd(nat,fun(nat,fun(nat,fun(nat,nat))),M2),N),R)),set_ord_atMost(nat,R)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),R) ).

% vandermonde
tff(fact_4016_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ).

% sum_gp_basic
tff(fact_4017_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A3: A,N: nat] :
          ( ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),C2),A3)),set_ord_atMost(nat,N)) = zero_zero(A) )
         => ~ ! [B4: fun(nat,A)] :
                ~ ! [Z4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,N)) = 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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),set_ord_lessThan(nat,N))) ) ) ).

% polyfun_linear_factor_root
tff(fact_4018_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),N: nat,A3: A] :
        ? [B4: fun(nat,A)] :
        ! [Z4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z4),A3)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),set_ord_lessThan(nat,N)))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),C2),A3)),set_ord_atMost(nat,N))) ) ).

% polyfun_linear_factor
tff(fact_4019_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M2: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))) ) ) ) ).

% sum_power_shift
tff(fact_4020_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_ff(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_ff(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_4021_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ff(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_ff(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_fh(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_4022_binomial,axiom,
    ! [A3: nat,B2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),N) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fi(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B2),N)),set_ord_atMost(nat,N)) ).

% binomial
tff(fact_4023_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cw(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% sum.in_pairs_0
tff(fact_4024_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M2: nat,A3: fun(nat,A),N: nat,B2: fun(nat,A),X: A] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),I3))
             => ( aa(nat,A,A3,I3) = zero_zero(A) ) )
         => ( ! [J: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J))
               => ( aa(nat,A,B2,J) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),A3),X)),set_ord_atMost(nat,M2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),B2),X)),set_ord_atMost(nat,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_fk(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A3),B2),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) ) ) ) ) ).

% polynomial_product
tff(fact_4025_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ay(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% prod.in_pairs_0
tff(fact_4026_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,K: A] :
          ( ! [X4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),set_ord_atMost(nat,N)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X4: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or1337092689740270186AtMost(nat,one_one(nat),N)))
               => ( aa(nat,A,C2,X4) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_4027_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fl(A,fun(nat,A),A3)),set_ord_atMost(nat,M2)) = 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))),M2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),M2)) ) ).

% gbinomial_sum_lower_neg
tff(fact_4028_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fm(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).

% binomial_ring
tff(fact_4029_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,B2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fn(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).

% pochhammer_binomial_sum
tff(fact_4030_polynomial__product__nat,axiom,
    ! [M2: nat,A3: fun(nat,nat),N: nat,B2: fun(nat,nat),X: nat] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),I3))
         => ( aa(nat,nat,A3,I3) = zero_zero(nat) ) )
     => ( ! [J: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J))
           => ( aa(nat,nat,B2,J) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fo(fun(nat,nat),fun(nat,fun(nat,nat)),A3),X)),set_ord_atMost(nat,M2))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fo(fun(nat,nat),fun(nat,fun(nat,nat)),B2),X)),set_ord_atMost(nat,N))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fq(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A3),B2),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_4031_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_ff(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_ff(fun(nat,A),fun(nat,real),B2))
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fh(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_4032_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,K6: real,C2: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),K6))
         => ( ! [X2: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X2)),K6))
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),C2),X2)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fr(A,fun(fun(nat,A),fun(nat,A)),X),C2)) ) ) ) ).

% termdiff_converges
tff(fact_4033_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(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_fs(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),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ft(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_4034_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_fu(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_fv(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_4035_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat,A3: A,X: A,Y2: A] : aa(set(nat),A,aa(fun(nat,A),fun(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_fw(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y2)),set_ord_atMost(nat,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(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_fx(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y2)),set_ord_atMost(nat,M2)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4036_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,Z: A,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N) = A3 )
          <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_fy(nat,fun(A,fun(A,fun(nat,A))),N),Z),A3)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_4037_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp0
tff(fact_4038_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( ( N != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fz(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_4039_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ga(nat,fun(nat,A),M2)),set_ord_atMost(nat,M2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2) ) ).

% gbinomial_sum_nat_pow2
tff(fact_4040_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat,A3: A,X: A,Y2: A] : aa(set(nat),A,aa(fun(nat,A),fun(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_fw(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y2)),set_ord_atMost(nat,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(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_gb(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y2)),set_ord_atMost(nat,M2)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_4041_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A3: fun(nat,A),X: A,Y2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),A3),X)),set_ord_atMost(nat,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),A3),Y2)),set_ord_atMost(nat,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)),aa(set(nat),A,aa(fun(nat,A),fun(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_gd(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A3),X),Y2)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff_alt
tff(fact_4042_binomial__r__part__sum,axiom,
    ! [M2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(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))),M2)),one_one(nat)))),set_ord_atMost(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),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2)) ).

% binomial_r_part_sum
tff(fact_4043_choose__linear__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ge(nat,fun(nat,nat),N)),set_ord_atMost(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).

% choose_linear_sum
tff(fact_4044_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gf(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_4045_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E2: real,C2: fun(nat,A),N: nat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
         => ? [M9: real] :
            ! [Z4: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),M9),real_V7770717601297561774m_norm(A,Z4)))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),E2),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z4)),aa(nat,nat,suc,N))))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_4046_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A3: fun(nat,A),X: A,Y2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),A3),X)),set_ord_atMost(nat,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),A3),Y2)),set_ord_atMost(nat,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)),aa(set(nat),A,aa(fun(nat,A),fun(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_gh(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A3),X),Y2)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff
tff(fact_4047_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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),M2))),one_one(A)))),set_ord_atMost(nat,M2)) = 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))),M2)) ) ).

% gbinomial_r_part_sum
tff(fact_4048_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gi(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_4049_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_gk(A,fun(A,fun(nat,A)),X),Y2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),X)),aa(A,A,sin(A),Y2))) ) ).

% sin_x_sin_y
tff(fact_4050_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_gm(A,fun(A,fun(nat,A)),X),Y2),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2))) ) ).

% sums_cos_x_plus_y
tff(fact_4051_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y2: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_go(A,fun(A,fun(nat,A)),X),Y2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),X)),aa(A,A,cos(A),Y2))) ) ).

% cos_x_cos_y
tff(fact_4052_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),X)),suminf(A,aTP_Lamp_gp(A,fun(nat,A),X))) ) ).

% exp_first_two_terms
tff(fact_4053_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real] : real_V8093663219630862766scaleR(A,A3,zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_4054_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A,B2: real] :
          ( ( real_V8093663219630862766scaleR(A,A3,X) = real_V8093663219630862766scaleR(A,B2,X) )
        <=> ( ( A3 = B2 )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_4055_scaleR__minus__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A] : real_V8093663219630862766scaleR(A,A3,aa(A,A,uminus_uminus(A),X)) = aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,A3,X)) ) ).

% scaleR_minus_right
tff(fact_4056_scaleR__one,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : real_V8093663219630862766scaleR(A,one_one(real),X) = X ) ).

% scaleR_one
tff(fact_4057_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A] :
          ( ( real_V8093663219630862766scaleR(A,A3,X) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(real) )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_4058_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : real_V8093663219630862766scaleR(A,zero_zero(real),X) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_4059_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),real_V8093663219630862766scaleR(A,U,A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),real_V8093663219630862766scaleR(A,U,B2)) )
        <=> ( ( A3 = B2 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_4060_scaleR__left_Ominus,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: real,Xa: A] : real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),X),Xa) = aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,X,Xa)) ) ).

% scaleR_left.minus
tff(fact_4061_scaleR__minus__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A] : real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),A3),X) = aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,A3,X)) ) ).

% scaleR_minus_left
tff(fact_4062_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),one_one(real)),X) = aa(A,A,uminus_uminus(A),X) ) ).

% scaleR_minus1_left
tff(fact_4063_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),U),A3)),real_V8093663219630862766scaleR(A,U,A3)) = A3 ) ).

% scaleR_collapse
tff(fact_4064_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V2: num,W: num,A3: A] : real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = 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),V2)),A3) ) ).

% inverse_scaleR_times
tff(fact_4065_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V2: num,W: num,A3: 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),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = 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),V2)),A3) ) ).

% fraction_scaleR_times
tff(fact_4066_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: 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_4067_scaleR__right__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A,Y2: A] : real_V8093663219630862766scaleR(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,A3,X)),real_V8093663219630862766scaleR(A,A3,Y2)) ) ).

% scaleR_right_diff_distrib
tff(fact_4068_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,A3: real,B2: real] :
          ( ( X != zero_zero(A) )
         => ( ( real_V8093663219630862766scaleR(A,A3,X) = real_V8093663219630862766scaleR(A,B2,X) )
           => ( A3 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_4069_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R: real] : real_Vector_of_real(A,R) = real_V8093663219630862766scaleR(A,R,one_one(A)) ) ).

% of_real_def
tff(fact_4070_scaleR__left_Odiff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: real,Y2: real,Xa: A] : real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y2),Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,X,Xa)),real_V8093663219630862766scaleR(A,Y2,Xa)) ) ).

% scaleR_left.diff
tff(fact_4071_scaleR__left__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,B2: real,X: A] : real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),B2),X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,A3,X)),real_V8093663219630862766scaleR(A,B2,X)) ) ).

% scaleR_left_diff_distrib
tff(fact_4072_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A3: real,C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,A3,C2)),real_V8093663219630862766scaleR(A,B2,C2))) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_4073_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,A3,X)),real_V8093663219630862766scaleR(A,B2,X))) ) ) ) ).

% scaleR_right_mono
tff(fact_4074_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V2: real,A3: A,X: A] :
          ( ( real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V2),A3) = X )
        <=> ( ( ( V2 = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V2 != zero_zero(real) )
             => ( real_V8093663219630862766scaleR(A,U,A3) = real_V8093663219630862766scaleR(A,V2,X) ) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_4075_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,U: real,V2: real,A3: A] :
          ( ( X = real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V2),A3) )
        <=> ( ( ( V2 = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V2 != zero_zero(real) )
             => ( real_V8093663219630862766scaleR(A,V2,X) = real_V8093663219630862766scaleR(A,U,A3) ) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_4076_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E2: A,C2: A,B2: real,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A3,E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,B2,E2)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),B2),E2)),C2)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_4077_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E2: A,C2: A,B2: real,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A3,E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,B2,E2)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3),E2)),D2))) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_4078_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),real_V8093663219630862766scaleR(A,A3,B2)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_4079_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,A3,B2)),zero_zero(A)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_4080_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),real_V8093663219630862766scaleR(A,A3,B2))) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_4081_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,A3,X)),zero_zero(A))) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_4082_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,A3,X)),zero_zero(A))) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_4083_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),real_V8093663219630862766scaleR(A,A3,X))) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_4084_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),real_V8093663219630862766scaleR(A,A3,B2))) ) ) ).

% split_scaleR_pos_le
tff(fact_4085_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,A3,X)),zero_zero(A))) ) ) ).

% split_scaleR_neg_le
tff(fact_4086_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: real,C2: A,D2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,A3,C2)),real_V8093663219630862766scaleR(A,B2,D2))) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_4087_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: real,X: A,Y2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,A3,X)),real_V8093663219630862766scaleR(A,B2,Y2))) ) ) ) ) ) ).

% scaleR_mono
tff(fact_4088_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,A3: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),one_one(real)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,A3,X)),X)) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_4089_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M2: real,X: A,C2: A,Y2: A] :
          ( ( M2 != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,M2,X)),C2) = Y2 )
          <=> ( X = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M2),Y2)),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M2),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_4090_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M2: real,Y2: A,X: A,C2: A] :
          ( ( M2 != zero_zero(real) )
         => ( ( Y2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,M2,X)),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M2),Y2)),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M2),C2)) = X ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_4091_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),real_V8093663219630862766scaleR(A,C2,A3))) ) ) ) ).

% neg_less_divideR_eq
tff(fact_4092_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),real_V8093663219630862766scaleR(A,C2,A3)),B2)) ) ) ) ).

% neg_divideR_less_eq
tff(fact_4093_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),real_V8093663219630862766scaleR(A,C2,A3)),B2)) ) ) ) ).

% pos_less_divideR_eq
tff(fact_4094_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),real_V8093663219630862766scaleR(A,C2,A3))) ) ) ) ).

% pos_divideR_less_eq
tff(fact_4095_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A3: real,X: A] :
          ( ( A3 != zero_zero(real) )
         => ( ( X != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),real_V8093663219630862766scaleR(A,A3,X)) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A3),aa(A,A,inverse_inverse(A),X)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_4096_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(A,aTP_Lamp_gq(A,fun(nat,A),X)) ) ).

% summable_exp_generic
tff(fact_4097_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_gr(A,fun(nat,A),X),aa(A,A,sin(A),X)) ) ).

% sin_converges
tff(fact_4098_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : aa(A,A,sin(A),X3) = suminf(A,aTP_Lamp_gr(A,fun(nat,A),X3)) ) ).

% sin_def
tff(fact_4099_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_gs(A,fun(nat,A),X),aa(A,A,cos(A),X)) ) ).

% cos_converges
tff(fact_4100_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : aa(A,A,cos(A),X3) = suminf(A,aTP_Lamp_gs(A,fun(nat,A),X3)) ) ).

% cos_def
tff(fact_4101_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_gt(A,fun(nat,real),X)) ) ).

% summable_norm_sin
tff(fact_4102_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_gu(A,fun(nat,real),X)) ) ).

% summable_norm_cos
tff(fact_4103_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,C2,A3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_4104_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C2,A3))) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_4105_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C2,A3))) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_4106_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),real_V8093663219630862766scaleR(A,C2,A3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_4107_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),real_V8093663219630862766scaleR(A,C2,A3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_4108_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C2,A3))) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_4109_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C2,A3))) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_4110_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),real_V8093663219630862766scaleR(A,C2,A3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_4111_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_gq(A,fun(nat,A),X),aa(A,A,exp(A),X)) ) ).

% exp_converges
tff(fact_4112_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : aa(A,A,exp(A),X3) = suminf(A,aTP_Lamp_gq(A,fun(nat,A),X3)) ) ).

% exp_def
tff(fact_4113_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_gv(A,fun(nat,real),X)) ) ).

% summable_norm_exp
tff(fact_4114_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_gw(A,fun(nat,A),X),aa(A,A,sin(A),X)) ) ).

% sin_minus_converges
tff(fact_4115_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_gx(A,fun(nat,A),X),aa(A,A,cos(A),X)) ) ).

% cos_minus_converges
tff(fact_4116_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,cosh(A),X) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% cosh_def
tff(fact_4117_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,sinh(A),X) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% sinh_def
tff(fact_4118_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y2) = aa(A,A,aa(A,fun(A,A),times_times(A),Y2),X) )
         => ( real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gy(A,fun(A,fun(nat,fun(nat,A))),X),Y2),N)),set_ord_atMost(nat,N)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4119_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_gz(A,fun(nat,A),X))) ) ).

% exp_first_term
tff(fact_4120_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,K: nat] : aa(A,A,exp(A),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gq(A,fun(nat,A),X)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ha(A,fun(nat,fun(nat,A)),X),K))) ) ).

% exp_first_terms
tff(fact_4121_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_hb(A,fun(nat,A),X),aa(A,A,cosh(A),X)) ) ).

% cosh_converges
tff(fact_4122_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_hc(A,fun(nat,A),X),aa(A,A,sinh(A),X)) ) ).

% sinh_converges
tff(fact_4123_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N5)),aa(nat,A,X6,aa(nat,nat,suc,N5))))
            | ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N5))),aa(nat,A,X6,N5))) ) ) ) ).

% monoseq_Suc
tff(fact_4124_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N2))),aa(nat,A,X6,N2)))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI2
tff(fact_4125_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,X6,aa(nat,nat,suc,N2))))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI1
tff(fact_4126_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_hd(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_4127_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),N: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N),I) = semiri8178284476397505188at_aux(A,Inc,N,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_4128_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_4129_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: fun(nat,A)] :
          ( topological_monoseq(A,A3)
         => topological_monoseq(A,aTP_Lamp_he(fun(nat,A),fun(nat,A),A3)) ) ) ).

% monoseq_minus
tff(fact_4130_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero(complex) )
       => ( arg(Z) = zero_zero(real) ) )
      & ( ( Z != zero_zero(complex) )
       => ( arg(Z) = fChoice(real,aTP_Lamp_hf(complex,fun(real,bool),Z)) ) ) ) ).

% Arg_def
tff(fact_4131_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y2: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y2 )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y2 = vEBT_Leaf(fFalse,fFalse) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y2 = vEBT_Leaf(fFalse,fFalse) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
                       => ( Y2 = 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,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_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) )
                      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
                       => ( Y2 = 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,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_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_4132_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M2),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_hg(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_4133_arctan__def,axiom,
    ! [Y2: real] : aa(real,real,arctan,Y2) = the(real,aTP_Lamp_hh(real,fun(real,bool),Y2)) ).

% arctan_def
tff(fact_4134_some__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X2: A] :
            ( pp(aa(A,bool,P,X2))
           => ( X2 = A3 ) )
       => ( fChoice(A,P) = A3 ) ) ) ).

% some_equality
tff(fact_4135_some__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aTP_Lamp_hi(A,fun(A,bool),X)) = X ).

% some_eq_trivial
tff(fact_4136_some__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aa(A,fun(A,bool),fequal(A),X)) = X ).

% some_sym_eq_trivial
tff(fact_4137_some__eq__imp,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A,B2: A] :
      ( ( fChoice(A,P) = A3 )
     => ( pp(aa(A,bool,P,B2))
       => pp(aa(A,bool,P,A3)) ) ) ).

% some_eq_imp
tff(fact_4138_tfl__some,axiom,
    ! [A: $tType,P5: fun(A,bool),X3: A] :
      ( pp(aa(A,bool,P5,X3))
     => pp(aa(A,bool,P5,fChoice(A,P5))) ) ).

% tfl_some
tff(fact_4139_Eps__cong,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X2: A] :
          ( pp(aa(A,bool,P,X2))
        <=> pp(aa(A,bool,Q,X2)) )
     => ( fChoice(A,P) = fChoice(A,Q) ) ) ).

% Eps_cong
tff(fact_4140_someI,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( pp(aa(A,bool,P,X))
     => pp(aa(A,bool,P,fChoice(A,P))) ) ).

% someI
tff(fact_4141_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,bool),A4: bool] :
      ( ( pp(aa(A,bool,P,fChoice(A,P)))
      <=> pp(A4) )
     => ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
      <=> pp(A4) ) ) ).

% verit_sko_ex'
tff(fact_4142_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ! [X_13: A] : pp(aa(A,bool,P,X_13))
    <=> pp(aa(A,bool,P,fChoice(A,aTP_Lamp_ab(fun(A,bool),fun(A,bool),P)))) ) ).

% verit_sko_forall
tff(fact_4143_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,bool),A4: bool] :
      ( ( pp(aa(A,bool,P,fChoice(A,aTP_Lamp_ab(fun(A,bool),fun(A,bool),P))))
      <=> pp(A4) )
     => ( ! [X_13: A] : pp(aa(A,bool,P,X_13))
      <=> pp(A4) ) ) ).

% verit_sko_forall'
tff(fact_4144_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B5: A,A4: A,P: fun(A,bool)] :
      ( ( B5 = A4 )
     => ( ( fChoice(A,P) = A4 )
      <=> ( fChoice(A,P) = B5 ) ) ) ).

% verit_sko_forall''
tff(fact_4145_verit__sko__ex__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,bool)] :
      ( ( X = fChoice(A,P) )
     => ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
      <=> pp(aa(A,bool,P,X)) ) ) ).

% verit_sko_ex_indirect
tff(fact_4146_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),P3: fun(A,bool)] :
      ( ( X = fChoice(A,P) )
     => ( ! [X2: A] :
            ( pp(aa(A,bool,P,X2))
          <=> pp(aa(A,bool,P3,X2)) )
       => ( ? [X_13: A] : pp(aa(A,bool,P3,X_13))
        <=> pp(aa(A,bool,P,X)) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_4147_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_ab(fun(A,bool),fun(A,bool),P)) )
     => ( ! [X_13: A] : pp(aa(A,bool,P,X_13))
      <=> pp(aa(A,bool,P,X)) ) ) ).

% verit_sko_forall_indirect
tff(fact_4148_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),P3: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_ab(fun(A,bool),fun(A,bool),P)) )
     => ( ! [X2: A] :
            ( pp(aa(A,bool,P,X2))
          <=> pp(aa(A,bool,P3,X2)) )
       => ( ! [X_13: A] : pp(aa(A,bool,P3,X_13))
        <=> pp(aa(A,bool,P,X)) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_4149_someI2,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X2: A] :
            ( pp(aa(A,bool,P,X2))
           => pp(aa(A,bool,Q,X2)) )
       => pp(aa(A,bool,Q,fChoice(A,P))) ) ) ).

% someI2
tff(fact_4150_someI__ex,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
     => pp(aa(A,bool,P,fChoice(A,P))) ) ).

% someI_ex
tff(fact_4151_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
     => ( ! [X2: A] :
            ( pp(aa(A,bool,P,X2))
           => pp(aa(A,bool,Q,X2)) )
       => pp(aa(A,bool,Q,fChoice(A,P))) ) ) ).

% someI2_ex
tff(fact_4152_someI2__bex,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A4))
          & pp(aa(A,bool,P,X3)) )
     => ( ! [X2: A] :
            ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
              & pp(aa(A,bool,P,X2)) )
           => pp(aa(A,bool,Q,X2)) )
       => pp(aa(A,bool,Q,fChoice(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_hj(set(A),fun(fun(A,bool),fun(A,bool)),A4),P)))) ) ) ).

% someI2_bex
tff(fact_4153_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( pp(aa(A,bool,P,fChoice(A,P)))
    <=> ? [X_13: A] : pp(aa(A,bool,P,X_13)) ) ).

% some_eq_ex
tff(fact_4154_some1__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( ? [X3: A] :
          ( pp(aa(A,bool,P,X3))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => ( Y3 = X3 ) ) )
     => ( pp(aa(A,bool,P,A3))
       => ( fChoice(A,P) = A3 ) ) ) ).

% some1_equality
tff(fact_4155_some__in__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),A4))),A4))
    <=> ( A4 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_4156_ln__real__def,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_hk(real,fun(real,bool),X)) ).

% ln_real_def
tff(fact_4157_ln__neg__is__const,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_hl(real,bool)) ) ) ).

% ln_neg_is_const
tff(fact_4158_arccos__def,axiom,
    ! [Y2: real] : aa(real,real,arccos,Y2) = the(real,aTP_Lamp_hm(real,fun(real,bool),Y2)) ).

% arccos_def
tff(fact_4159_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_hn(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_4160_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ho(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_4161_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_hp(real,bool)) ).

% pi_half
tff(fact_4162_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_hp(real,bool))) ).

% pi_def
tff(fact_4163_divmod__step__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_hq(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_4164_arcsin__def,axiom,
    ! [Y2: real] : aa(real,real,arcsin,Y2) = the(real,aTP_Lamp_hr(real,fun(real,bool),Y2)) ).

% arcsin_def
tff(fact_4165_divmod__nat__if,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( ( N = zero_zero(nat) )
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) )
       => ( divmod_nat(M2,N) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),M2) ) )
      & ( ~ ( ( N = zero_zero(nat) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) )
       => ( divmod_nat(M2,N) = aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_hs(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_4166_set__encode__def,axiom,
    nat_set_encode = aa(fun(nat,nat),fun(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_encode_def
tff(fact_4167_Sum__Ico__nat,axiom,
    ! [M2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ai(nat,nat)),set_or7035219750837199246ssThan(nat,M2,N)) = 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),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Ico_nat
tff(fact_4168_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_4169_set__decode__inverse,axiom,
    ! [N: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(N)) = N ).

% set_decode_inverse
tff(fact_4170_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or7035219750837199246ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),U)) ) ) ) ).

% atLeastLessThan_iff
tff(fact_4171_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4172_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_4173_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,N: A,M2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),N))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I,M2)),set_or7035219750837199246ssThan(A,I,N)) = set_or7035219750837199246ssThan(A,N,M2) ) ) ) ).

% ivl_diff
tff(fact_4174_lessThan__minus__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: A,M2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_ord_lessThan(A,N)),set_ord_lessThan(A,M2)) = set_or7035219750837199246ssThan(A,M2,N) ) ).

% lessThan_minus_lessThan
tff(fact_4175_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_4176_atLeastLessThan__singleton,axiom,
    ! [M2: nat] : set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,M2)) = aa(set(nat),set(nat),insert(nat,M2),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_4177_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M2: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_4178_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M2: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = 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,M2,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_4179_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
            <=> ( ( A3 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4180_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) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
             => ( A3 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4181_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) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4182_all__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
         => pp(aa(nat,bool,P,M5)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X4)) ) ) ).

% all_nat_less_eq
tff(fact_4183_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
          & pp(aa(nat,bool,P,M5)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X4)) ) ) ).

% ex_nat_less_eq
tff(fact_4184_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,L,aa(nat,nat,suc,U)) = set_or1337092689740270186AtMost(nat,L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_4185_lessThan__atLeast0,axiom,
    ! [N: nat] : set_ord_lessThan(nat,N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).

% lessThan_atLeast0
tff(fact_4186_atLeastLessThan0,axiom,
    ! [M2: nat] : set_or7035219750837199246ssThan(nat,M2,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_4187_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_4188_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_at(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_4189_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_ht(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hv(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% sum.triangle_reindex_eq
tff(fact_4190_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: 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),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_ht(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hx(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% prod.triangle_reindex_eq
tff(fact_4191_sum_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_add(A) )
     => ! [A3: B,C2: B,B2: B,D2: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A3 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X2: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X2))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),D2))
                   => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_4192_prod_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_mult(A) )
     => ! [A3: B,C2: B,B2: B,D2: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A3 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X2: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X2))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),D2))
                   => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_4193_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,P2: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P2))
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M2,P2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M2,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,N,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_4194_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_hy(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hv(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.triangle_reindex
tff(fact_4195_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: 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),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_hy(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hx(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% prod.triangle_reindex
tff(fact_4196_size__list__estimation,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y2: nat,F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y2),aa(A,nat,F2,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y2),size_list(A,F2,Xs))) ) ) ).

% size_list_estimation
tff(fact_4197_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_4198_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4199_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4200_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_4201_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_4202_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_4203_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_4204_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] : set_or7035219750837199246ssThan(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4205_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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),N))),aa(nat,A,G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4206_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4207_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_4208_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cp(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,N)),aa(nat,A,F2,M2)) ) ) ) ).

% sum_Suc_diff'
tff(fact_4209_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hz(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M2)),set_or7035219750837199246ssThan(nat,N,M2)) ) ).

% sum.atLeastLessThan_rev
tff(fact_4210_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,N),set_or7035219750837199246ssThan(nat,M2,N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThanSuc
tff(fact_4211_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ia(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_es(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_4212_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M2)) = 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_ib(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M2)),set_or7035219750837199246ssThan(nat,N,M2)) ) ).

% prod.atLeastLessThan_rev
tff(fact_4213_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ic(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ev(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_4214_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_Suc_fact
tff(fact_4215_prod__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_fact
tff(fact_4216_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M2: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.head_if
tff(fact_4217_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M2: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = 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,M2,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.head_if
tff(fact_4218_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_Suc
tff(fact_4219_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_cn(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M2)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4220_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M2)) = 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_au(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M2)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4221_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aw(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_4222_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aa(nat,fun(nat,nat),minus_minus(nat),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_rev
tff(fact_4223_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A,K: nat] :
          ( sums(A,F2,S)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_id(fun(nat,A),fun(nat,fun(nat,A)),F2),K),S) ) ) ) ).

% sums_group
tff(fact_4224_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A] : bit_se2584673776208193580ke_bit(A,N,A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ie(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_4225_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_lessThan(nat,N)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_4226_divmod__nat__def,axiom,
    ! [M2: nat,N: nat] : divmod_nat(M2,N) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N)),modulo_modulo(nat,M2,N)) ).

% divmod_nat_def
tff(fact_4227_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),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),N),K),N)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).

% fact_split
tff(fact_4228_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_if(nat,fun(nat,fun(nat,A)),K),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_4229_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_ig(A,fun(nat,fun(nat,A)),A3),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_4230_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_ih(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_4231_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_ih(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_4232_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_az(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_4233_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(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_4234_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_4235_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_ii(nat,fun(nat,A))),divmod_nat(N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ) ).

% of_nat_code_if
tff(fact_4236_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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_ij(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_4237_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I3: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,I3)),aa(nat,A,A3,J))) ) )
         => ( ! [I3: nat,J: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,B2,J)),aa(nat,A,B2,I3))) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_4238_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A3: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I3: nat,J: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,A3,I3)),aa(nat,nat,A3,J))) ) )
     => ( ! [I3: nat,J: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,B2,J)),aa(nat,nat,B2,I3))) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_il(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A3),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_4239_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y2: B] : fChoice(product_prod(A,B),product_case_prod(A,B,bool,aa(B,fun(A,fun(B,bool)),aTP_Lamp_im(A,fun(B,fun(A,fun(B,bool))),X),Y2))) = aa(B,product_prod(A,B),product_Pair(A,B,X),Y2) ).

% Eps_case_prod_eq
tff(fact_4240_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),product_case_prod(A,B,bool,aTP_Lamp_in(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),P))) ).

% split_paired_Eps
tff(fact_4241_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,aa(int,fun(int,int),plus_plus(int),U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_4242_set__encode__insert,axiom,
    ! [A4: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A4))
     => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),A4))
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,N),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))),N)),aa(set(nat),nat,nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_4243_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiring_1_of_nat(int) ).

% int_of_nat_def
tff(fact_4244_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_4245_divmod__step__integer__def,axiom,
    ! [L: num,Qr: product_prod(code_integer,code_integer)] : unique1321980374590559556d_step(code_integer,L,Qr) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_io(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_4246_sum__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [F4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),F4))
         => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),F4) = zero_zero(A) )
          <=> ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),F4))
               => ( aa(B,A,F2,X4) = zero_zero(A) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4247_sum_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),G: fun(B,A)] :
          ( ~ pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = zero_zero(A) ) ) ) ).

% sum.infinite
tff(fact_4248_prod__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom(A)
     => ! [A4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4) = zero_zero(A) )
          <=> ? [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
                & ( aa(B,A,F2,X4) = zero_zero(A) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_4249_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),set_or1337092689740270186AtMost(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Icc_iff
tff(fact_4250_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),set_or7035219750837199246ssThan(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ico_iff
tff(fact_4251_prod_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),G: fun(B,A)] :
          ( ~ pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = one_one(A) ) ) ) ).

% prod.infinite
tff(fact_4252_sum_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ip(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ip(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = zero_zero(A) ) ) ) ) ) ).

% sum.delta
tff(fact_4253_sum_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_iq(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_iq(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = zero_zero(A) ) ) ) ) ) ).

% sum.delta'
tff(fact_4254_prod__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,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) )
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
           => ( aa(A,nat,F2,X4) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_4255_prod_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( 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_ir(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( 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_ir(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = one_one(A) ) ) ) ) ) ).

% prod.delta'
tff(fact_4256_prod_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( 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_is(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( 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_is(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = one_one(A) ) ) ) ) ) ).

% prod.delta
tff(fact_4257_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,bool),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),collect(nat,P)))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_it(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_4258_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A4: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),A4))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_iu(set(nat),fun(fun(nat,A),fun(nat,A)),A4),F2)) ) ) ).

% summable_If_finite_set
tff(fact_4259_set__encode__inverse,axiom,
    ! [A4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A4))
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A4)) = A4 ) ) ).

% set_encode_inverse
tff(fact_4260_prod__pos__nat__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A4)))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X4))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_4261_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A4: set(nat),C2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite(nat),A4))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_iv(fun(nat,A),fun(nat,A),C2)),A4) = aa(nat,A,C2,zero_zero(nat)) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite(nat),A4))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_iv(fun(nat,A),fun(nat,A),C2)),A4) = zero_zero(A) ) ) ) ) ).

% sum_zero_power
tff(fact_4262_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A4: set(nat),C2: fun(nat,A),D2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite(nat),A4))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_iw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),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))) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite(nat),A4))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_iw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A4) = zero_zero(A) ) ) ) ) ).

% sum_zero_power'
tff(fact_4263_set__encode__eq,axiom,
    ! [A4: set(nat),B5: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A4))
     => ( pp(aa(set(nat),bool,finite_finite(nat),B5))
       => ( ( aa(set(nat),nat,nat_set_encode,A4) = aa(set(nat),nat,nat_set_encode,B5) )
        <=> ( A4 = B5 ) ) ) ) ).

% set_encode_eq
tff(fact_4264_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),K),zero_zero(code_integer)) = K ).

% minus_integer_code(1)
tff(fact_4265_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),zero_zero(code_integer)),L) = aa(code_integer,code_integer,uminus_uminus(code_integer),L) ).

% minus_integer_code(2)
tff(fact_4266_divmod__integer_H__def,axiom,
    ! [M2: num,N: num] : unique8689654367752047608divmod(code_integer,M2,N) = aa(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),divide_divide(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M2)),aa(num,code_integer,numeral_numeral(code_integer),N))),modulo_modulo(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M2),aa(num,code_integer,numeral_numeral(code_integer),N))) ).

% divmod_integer'_def
tff(fact_4267_bounded__nat__set__is__finite,axiom,
    ! [N6: set(nat),N: nat] :
      ( ! [X2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),N6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),N)) )
     => pp(aa(set(nat),bool,finite_finite(nat),N6)) ) ).

% bounded_nat_set_is_finite
tff(fact_4268_finite__nat__set__iff__bounded,axiom,
    ! [N6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),N6))
    <=> ? [M5: nat] :
        ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),N6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),M5)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_4269_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = zero_zero(code_integer) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
           => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
           => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = one_one(code_integer) ) ) ) ) ) ).

% sgn_integer_code
tff(fact_4270_finite__set__decode,axiom,
    ! [N: nat] : pp(aa(set(nat),bool,finite_finite(nat),nat_set_decode(N))) ).

% finite_set_decode
tff(fact_4271_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,bool),I: nat] : pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ix(fun(nat,bool),fun(nat,fun(nat,bool)),P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_4272_finite__lists__length__eq,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_iy(set(A),fun(nat,fun(list(A),bool)),A4),N)))) ) ).

% finite_lists_length_eq
tff(fact_4273_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S3))
                & ~ ? [Xa2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa2),X2)) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4274_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X6: set(A)] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
               => ? [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),X6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Xa2)) ) )
           => ~ pp(aa(set(A),bool,finite_finite(A),X6)) ) ) ) ).

% infinite_growing
tff(fact_4275_prod__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [A4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( ? [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A4))
                & ( aa(B,A,F2,X3) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4) = zero_zero(A) ) ) ) ) ).

% prod_zero
tff(fact_4276_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite(A),set_or1337092689740270186AtMost(A,A3,B2))) ) ) ).

% infinite_Icc
tff(fact_4277_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite(A),set_or7035219750837199246ssThan(A,A3,B2))) ) ) ).

% infinite_Ico
tff(fact_4278_finite__lists__length__le,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_iz(set(A),fun(nat,fun(list(A),bool)),A4),N)))) ) ).

% finite_lists_length_le
tff(fact_4279_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N6: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),N6))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N6))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_4280_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),X: fun(B,A),Y2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),Y2))))
           => pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_jb(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y2)))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_4281_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),X: fun(B,A),Y2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,A),fun(B,bool)),I5),Y2))))
           => pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_jd(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y2)))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_4282_sum_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),G: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),collect(B,aa(fun(B,bool),fun(B,bool),aTP_Lamp_je(set(B),fun(fun(B,bool),fun(B,bool)),A4),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_jf(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A4) ) ) ) ).

% sum.inter_filter
tff(fact_4283_set__encode__inf,axiom,
    ! [A4: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite(nat),A4))
     => ( aa(set(nat),nat,nat_set_encode,A4) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_4284_prod_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),G: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),collect(B,aa(fun(B,bool),fun(B,bool),aTP_Lamp_je(set(B),fun(fun(B,bool),fun(B,bool)),A4),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_jg(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A4) ) ) ) ).

% prod.inter_filter
tff(fact_4285_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] : pp(aa(set(A),bool,finite_finite(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_jh(A,fun(A,fun(A,bool)),A3),B2)))) ) ).

% finite_int_segment
tff(fact_4286_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),T2: set(C),G: fun(C,A),I: fun(C,B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( pp(aa(set(C),bool,finite_finite(C),T2))
           => ( ! [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),T2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(C,A,G,X2))) )
             => ( ! [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S))
                   => ? [Xa2: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Xa2),T2))
                        & ( aa(C,B,I,Xa2) = X2 )
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X2)),aa(C,A,G,Xa2))) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),S)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T2))) ) ) ) ) ) ).

% sum_le_included
tff(fact_4287_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( ! [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X2))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4) = zero_zero(A) )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
                 => ( aa(B,A,F2,X4) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4288_sum__strict__mono__ex1,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A4: set(I6),F2: fun(I6,A),G: fun(I6,A)] :
          ( pp(aa(set(I6),bool,finite_finite(I6),A4))
         => ( ! [X2: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X2),A4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F2,X2)),aa(I6,A,G,X2))) )
           => ( ? [X3: I6] :
                  ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X3),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(I6,A,F2,X3)),aa(I6,A,G,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),F2),A4)),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),G),A4))) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4289_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R2: fun(A,fun(A,bool)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R2,zero_zero(A)),zero_zero(A)))
         => ( ! [X1: A,Y1: A,X24: A,Y24: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X1),X24))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R2,Y1),Y24)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X24),Y24))) )
           => ( pp(aa(set(B),bool,finite_finite(B),S3))
             => ( ! [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(B,A,H,X2)),aa(B,A,G,X2))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3))) ) ) ) ) ) ).

% sum.related
tff(fact_4290_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
           => ( ! [B4: A,A8: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite(A),A8))
                 => ( ! [X3: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A8))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),X3)) )
                   => ( pp(aa(set(A),bool,P,A8))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),insert(A,B4),A8))) ) ) )
             => pp(aa(set(A),bool,P,A4)) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4291_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
           => ( ! [B4: A,A8: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite(A),A8))
                 => ( ! [X3: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A8))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),B4)) )
                   => ( pp(aa(set(A),bool,P,A8))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),insert(A,B4),A8))) ) ) )
             => pp(aa(set(A),bool,P,A4)) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4292_sum__strict__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A4: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( ( A4 != bot_bot(set(B)) )
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X2)),aa(B,A,G,X2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4))) ) ) ) ) ).

% sum_strict_mono
tff(fact_4293_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R2: fun(A,fun(A,bool)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R2,one_one(A)),one_one(A)))
         => ( ! [X1: A,Y1: A,X24: A,Y24: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X1),X24))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R2,Y1),Y24)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X24),Y24))) )
           => ( pp(aa(set(B),bool,finite_finite(B),S3))
             => ( ! [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(B,A,H,X2)),aa(B,A,G,X2))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),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_4294_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S4: set(B),T4: set(C),S3: set(B),I: fun(C,B),J2: fun(B,C),T5: set(C),G: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S4))
         => ( pp(aa(set(C),bool,finite_finite(C),T4))
           => ( ! [A6: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S4)))
                 => ( aa(C,B,I,aa(B,C,J2,A6)) = A6 ) )
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S4)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J2,A6)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4)))
                     => ( aa(B,C,J2,aa(C,B,I,B4)) = B4 ) )
                 => ( ! [B4: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I,B4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S4))) )
                   => ( ! [A6: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),S4))
                         => ( aa(B,A,G,A6) = zero_zero(A) ) )
                     => ( ! [B4: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T4))
                           => ( aa(C,A,H,B4) = zero_zero(A) ) )
                       => ( ! [A6: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),S3))
                             => ( aa(C,A,H,aa(B,C,J2,A6)) = aa(B,A,G,A6) ) )
                         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),H),T5) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4295_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S4: set(B),T4: set(C),S3: set(B),I: fun(C,B),J2: fun(B,C),T5: set(C),G: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S4))
         => ( pp(aa(set(C),bool,finite_finite(C),T4))
           => ( ! [A6: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S4)))
                 => ( aa(C,B,I,aa(B,C,J2,A6)) = A6 ) )
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S4)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J2,A6)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4)))
                     => ( aa(B,C,J2,aa(C,B,I,B4)) = B4 ) )
                 => ( ! [B4: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I,B4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S4))) )
                   => ( ! [A6: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),S4))
                         => ( aa(B,A,G,A6) = one_one(A) ) )
                     => ( ! [B4: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T4))
                           => ( aa(C,A,H,B4) = one_one(A) ) )
                       => ( ! [A6: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),S3))
                             => ( aa(C,A,H,aa(B,C,J2,A6)) = aa(B,A,G,A6) ) )
                         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),H),T5) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_4296_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
            & ( aa(A,nat,F2,X4) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A4))
               => ( ( X4 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4297_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_4298_sum__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4) = one_one(nat) )
      <=> ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
            & ( aa(A,nat,F2,X4) = one_one(nat) )
            & ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A4))
               => ( ( X4 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4299_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_4300_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),F2: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I3))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),S) = zero_zero(A) )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),S))
               => ( aa(B,A,F2,I) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4301_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),F2: fun(B,A),B5: A,I: B] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I3))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),S) = B5 )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),B5)) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4302_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_4303_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_4304_sum_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),collect(B,aTP_Lamp_ji(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4305_prod_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),collect(B,aTP_Lamp_jj(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_4306_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aTP_Lamp_jk(nat,fun(nat,bool),M2)))) ) ).

% finite_divisors_nat
tff(fact_4307_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N6: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),N6))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N6))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => sums(A,F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),N6)) ) ) ) ).

% sums_finite
tff(fact_4308_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,bool),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),collect(nat,P)))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_it(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),P),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),collect(nat,P))) ) ) ).

% sums_If_finite
tff(fact_4309_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A4: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),A4))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_iu(set(nat),fun(fun(nat,A),fun(nat,A)),A4),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A4)) ) ) ).

% sums_If_finite_set
tff(fact_4310_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N6: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),N6))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N6))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => ( suminf(A,F2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),N6) ) ) ) ) ).

% suminf_finite
tff(fact_4311_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A] : pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_jl(A,fun(A,bool),A3)))) ) ).

% finite_abs_int_segment
tff(fact_4312_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N6: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N6),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite(nat),N6)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4313_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite(nat),N6)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_4314_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),I5))
         => ( aa(B,B,exp(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aTP_Lamp_jm(fun(A,B),fun(A,B),F2)),I5) ) ) ) ).

% exp_sum
tff(fact_4315_sum__pos2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),I: B,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),I5))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F2,I)))
             => ( ! [I3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),I5))) ) ) ) ) ) ).

% sum_pos2
tff(fact_4316_sum__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),I5))
         => ( ( I5 != bot_bot(set(B)) )
           => ( ! [I3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F2,I3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),I5))) ) ) ) ) ).

% sum_pos
tff(fact_4317_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),I5))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
           => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F2,I)))
             => ( ! [I3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F2,I3))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),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_4318_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),I5))
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F2,I3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),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_4319_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C5: set(B),A4: set(B),B5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C5))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),A4)))
                   => ( aa(B,A,G,A6) = zero_zero(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),B5)))
                     => ( aa(B,A,H,B4) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B5) )
                  <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C5) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4320_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C5: set(B),A4: set(B),B5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C5))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),A4)))
                   => ( aa(B,A,G,A6) = zero_zero(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),B5)))
                     => ( aa(B,A,H,B4) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C5) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B5) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4321_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,G,X2) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T5) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4322_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,G,X2) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4323_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,H,X2) = zero_zero(A) ) )
             => ( ! [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                   => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T5) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4324_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,G,X2) = zero_zero(A) ) )
             => ( ! [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                   => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4325_sum_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B5: set(B),A4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A4))
         => ( pp(aa(set(B),bool,finite_finite(B),A4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B5)) ) ) ) ) ).

% sum.subset_diff
tff(fact_4326_sum__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A4: set(B),B5: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B5)) ) ) ) ) ).

% sum_diff
tff(fact_4327_prod_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B5: set(B),A4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A4))
         => ( pp(aa(set(B),bool,finite_finite(B),A4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B5)) ) ) ) ) ).

% prod.subset_diff
tff(fact_4328_prod_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T5: set(B),S3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,G,X2) = one_one(A) ) )
             => ( ! [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                   => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_4329_prod_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T5: set(B),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,H,X2) = one_one(A) ) )
             => ( ! [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                   => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T5) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_4330_prod_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T5: set(B),S3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,G,X2) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_4331_prod_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T5: set(B),S3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,G,X2) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T5) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_4332_prod_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C5: set(B),A4: set(B),B5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C5))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),A4)))
                   => ( aa(B,A,G,A6) = one_one(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),B5)))
                     => ( aa(B,A,H,B4) = one_one(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C5) )
                   => ( 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),B5) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_4333_prod_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C5: set(B),A4: set(B),B5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C5))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),A4)))
                   => ( aa(B,A,G,A6) = one_one(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),B5)))
                     => ( aa(B,A,H,B4) = one_one(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),B5) )
                  <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_4334_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
     => ? [H2: fun(A,A)] : bij_betw(A,A,H2,A4,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))))) ) ).

% infinite_imp_bij_betw
tff(fact_4335_sum__diff__nat,axiom,
    ! [A: $tType,B5: set(A),A4: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A4))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B5)) ) ) ) ).

% sum_diff_nat
tff(fact_4336_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_jn(nat,fun(A,bool),N)))) ) ) ).

% finite_roots_unity
tff(fact_4337_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S4: set(B),T4: set(C),H: fun(B,C),S3: set(B),T5: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S4))
         => ( pp(aa(set(C),bool,finite_finite(C),T4))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),S4))
                   => ( aa(C,A,G,aa(B,C,H,A6)) = zero_zero(A) ) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T4))
                     => ( aa(C,A,G,B4) = zero_zero(A) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_jo(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T5) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_4338_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S3: A,A4: set(nat),S4: A,F2: fun(nat,A)] :
          ( sums(A,G,S3)
         => ( pp(aa(set(nat),bool,finite_finite(nat),A4))
           => ( ( S4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_jp(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_jq(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A4),F2),S4) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_4339_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S4: set(B),T4: set(C),H: fun(B,C),S3: set(B),T5: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S4))
         => ( pp(aa(set(C),bool,finite_finite(C),T4))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A6),S4))
                   => ( aa(C,A,G,aa(B,C,H,A6)) = one_one(A) ) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T4))
                     => ( aa(C,A,G,B4) = one_one(A) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_jr(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),T5) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_4340_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( pp(aa(set(A),bool,finite_finite(A),I5))
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,I3))) )
       => ( aa(real,real,ln_ln(real),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),F2),I5)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_js(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_4341_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A4: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I3)))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,I3)),aa(B,A,G,I3))) ) )
           => ( ( A4 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(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_mono_strict
tff(fact_4342_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [B5: set(B),A4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),B5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),B5))
           => ( ! [B4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A4)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,B4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B5))) ) ) ) ) ).

% sum_mono2
tff(fact_4343_sum_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),G: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),insert(B,X),A4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ).

% sum.insert_remove
tff(fact_4344_sum_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% sum.remove
tff(fact_4345_sum__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A4: set(B),A3: B,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),A4))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),aa(B,A,F2,A3)) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),A4))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4) ) ) ) ) ) ).

% sum_diff1
tff(fact_4346_prod_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% prod.remove
tff(fact_4347_prod_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),G: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),insert(B,X),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ).

% prod.insert_remove
tff(fact_4348_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I5: set(nat)] :
          ( summable(A,F2)
         => ( pp(aa(set(nat),bool,finite_finite(nat),I5))
           => ( ! [N2: nat] :
                  ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),aa(set(nat),set(nat),uminus_uminus(set(nat)),I5)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),I5)),suminf(A,F2))) ) ) ) ) ).

% sum_le_suminf
tff(fact_4349_sum_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_jt(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),aa(set(B),set(B),insert(B,A3),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_jt(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))) ) ) ) ) ) ).

% sum.delta_remove
tff(fact_4350_prod_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ju(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),aa(set(B),set(B),insert(B,A3),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ju(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))) ) ) ) ) ) ).

% prod.delta_remove
tff(fact_4351_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B5: set(A),A4: set(A),B2: A,F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),B5))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)))
             => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),zero_zero(B)),aa(A,B,F2,B2)))
               => ( ! [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B5))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2))) )
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B5))) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4352_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: C,A4: set(C),F2: fun(C,B)] :
          ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I),A4))
         => ( ! [X2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),A4),aa(set(C),set(C),insert(C,I),bot_bot(set(C))))))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(C,B,F2,X2))) )
           => ( pp(aa(set(C),bool,finite_finite(C),A4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F2,I)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F2),A4))) ) ) ) ) ).

% member_le_sum
tff(fact_4353_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B5: set(A),A4: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),B5))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F2,B4))) )
             => ( ! [A6: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),A4))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A6))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),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),B5))) ) ) ) ) ) ).

% prod_mono2
tff(fact_4354_prod__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom_divide(A)
     => ! [A4: set(B),F2: fun(B,A),A3: B] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( ( aa(B,A,F2,A3) != zero_zero(A) )
           => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),A4))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))) = 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(B,A,F2,A3)) ) )
              & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),A4))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4) ) ) ) ) ) ) ).

% prod_diff1
tff(fact_4355_even__set__encode__iff,axiom,
    ! [A4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(nat),nat,nat_set_encode,A4)))
      <=> ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4)) ) ) ).

% even_set_encode_iff
tff(fact_4356_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => pp(aa(set(A),bool,finite_finite(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jv(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))) ) ) ) ).

% polyfun_roots_finite
tff(fact_4357_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jv(fun(nat,A),fun(nat,fun(A,bool)),C2),N))))
        <=> ? [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
              & ( aa(nat,A,C2,I2) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_4358_finite__Diff__insert,axiom,
    ! [A: $tType,A4: set(A),A3: A,B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(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),B5))))
    <=> pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5))) ) ).

% finite_Diff_insert
tff(fact_4359_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),K)))) ).

% finite_Collect_less_nat
tff(fact_4360_finite__induct__select,axiom,
    ! [A: $tType,S3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),S3))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [T6: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),T6),S3))
             => ( pp(aa(set(A),bool,P,T6))
               => ? [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T6)))
                    & pp(aa(set(A),bool,P,aa(set(A),set(A),insert(A,X3),T6))) ) ) )
         => pp(aa(set(A),bool,P,S3)) ) ) ) ).

% finite_induct_select
tff(fact_4361_finite__Diff2,axiom,
    ! [A: $tType,B5: set(A),A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B5))
     => ( pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)))
      <=> pp(aa(set(A),bool,finite_finite(A),A4)) ) ) ).

% finite_Diff2
tff(fact_4362_finite__interval__int1,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_jw(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int1
tff(fact_4363_finite__interval__int4,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_jx(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int4
tff(fact_4364_finite__Diff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5))) ) ).

% finite_Diff
tff(fact_4365_finite__interval__int2,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_jy(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int2
tff(fact_4366_finite__interval__int3,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_jz(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int3
tff(fact_4367_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(set(complex),bool,finite_finite(complex),collect(complex,aa(complex,fun(complex,bool),aTP_Lamp_di(nat,fun(complex,fun(complex,bool)),N),C2)))) ) ).

% finite_nth_roots
tff(fact_4368_uminus__integer__code_I1_J,axiom,
    aa(code_integer,code_integer,uminus_uminus(code_integer),zero_zero(code_integer)) = zero_zero(code_integer) ).

% uminus_integer_code(1)
tff(fact_4369_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),K) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = K ) ) ) ).

% abs_integer_code
tff(fact_4370_finite__maxlen,axiom,
    ! [A: $tType,M10: set(list(A))] :
      ( pp(aa(set(list(A)),bool,finite_finite(list(A)),M10))
     => ? [N2: nat] :
        ! [X3: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X3),M10))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X3)),N2)) ) ) ).

% finite_maxlen
tff(fact_4371_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa(set(int),bool,finite_finite(int),set_or7035219750837199246ssThan(int,zero_zero(int),U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_4372_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => pp(aa(set(int),bool,finite_finite(int),collect(int,aTP_Lamp_ka(int,fun(int,bool),I)))) ) ).

% finite_divisors_int
tff(fact_4373_Diff__infinite__finite,axiom,
    ! [A: $tType,T5: set(A),S3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),T5))
     => ( ~ pp(aa(set(A),bool,finite_finite(A),S3))
       => ~ pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T5))) ) ) ).

% Diff_infinite_finite
tff(fact_4374_finite__empty__induct,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,P,A4))
       => ( ! [A6: A,A8: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),A8))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),A8))
               => ( pp(aa(set(A),bool,P,A8))
                 => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),insert(A,A6),bot_bot(set(A)))))) ) ) )
         => pp(aa(set(A),bool,P,bot_bot(set(A)))) ) ) ) ).

% finite_empty_induct
tff(fact_4375_infinite__coinduct,axiom,
    ! [A: $tType,X6: fun(set(A),bool),A4: set(A)] :
      ( pp(aa(set(A),bool,X6,A4))
     => ( ! [A8: set(A)] :
            ( pp(aa(set(A),bool,X6,A8))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A8))
                & ( pp(aa(set(A),bool,X6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),insert(A,X3),bot_bot(set(A))))))
                  | ~ pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),insert(A,X3),bot_bot(set(A)))))) ) ) )
       => ~ pp(aa(set(A),bool,finite_finite(A),A4)) ) ) ).

% infinite_coinduct
tff(fact_4376_infinite__remove,axiom,
    ! [A: $tType,S3: set(A),A3: A] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),S3))
     => ~ pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) ) ).

% infinite_remove
tff(fact_4377_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),bool),B5: set(A)] :
      ( pp(aa(set(A),bool,P,bot_bot(set(A))))
     => ( ( ~ pp(aa(set(A),bool,finite_finite(A),B5))
         => pp(aa(set(A),bool,P,B5)) )
       => ( ! [A8: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),A8))
             => ( ( A8 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),B5))
                 => ( ! [X3: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A8))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),insert(A,X3),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A8)) ) ) ) )
         => pp(aa(set(A),bool,P,B5)) ) ) ) ).

% remove_induct
tff(fact_4378_finite__remove__induct,axiom,
    ! [A: $tType,B5: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),B5))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [A8: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),A8))
             => ( ( A8 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),B5))
                 => ( ! [X3: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A8))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),insert(A,X3),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A8)) ) ) ) )
         => pp(aa(set(A),bool,P,B5)) ) ) ) ).

% finite_remove_induct
tff(fact_4379_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( aa(int,code_integer,code_integer_of_int,K) = aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),K))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( ( ( K = zero_zero(int) )
           => ( aa(int,code_integer,code_integer_of_int,K) = zero_zero(code_integer) ) )
          & ( ( K != zero_zero(int) )
           => ( aa(int,code_integer,code_integer_of_int,K) = if(code_integer,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),aa(int,code_integer,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))))),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))),aa(int,code_integer,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)))))),one_one(code_integer))) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_4380_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ! [X2: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
         => ( aa(list(A),nat,size_size(list(A)),X2) = N ) )
     => ( aa(list(A),nat,size_size(list(A)),concat(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(list(A)),nat,size_size(list(list(A))),Xs)),N) ) ) ).

% length_mul_elem
tff(fact_4381_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collect(product_prod(int,int),product_case_prod(int,int,bool,aTP_Lamp_kb(int,fun(int,fun(int,bool)),D2))) ).

% int_ge_less_than2_def
tff(fact_4382_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collect(product_prod(int,int),product_case_prod(int,int,bool,aTP_Lamp_kc(int,fun(int,fun(int,bool)),D2))) ).

% int_ge_less_than_def
tff(fact_4383_zero__integer__def,axiom,
    zero_zero(code_integer) = aa(int,code_integer,code_integer_of_int,zero_zero(int)) ).

% zero_integer_def
tff(fact_4384_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Xa),X)) ) ).

% less_integer.abs_eq
tff(fact_4385_uminus__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),X)) ).

% uminus_integer.abs_eq
tff(fact_4386_one__integer__def,axiom,
    one_one(code_integer) = aa(int,code_integer,code_integer_of_int,one_one(int)) ).

% one_integer_def
tff(fact_4387_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),X)) ).

% minus_integer.abs_eq
tff(fact_4388_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),X)) ).

% divide_integer.abs_eq
tff(fact_4389_infinite__int__iff__unbounded,axiom,
    ! [S3: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite(int),S3))
    <=> ! [M5: int] :
        ? [N5: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M5),aa(int,int,abs_abs(int),N5)))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),N5),S3)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_4390_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite(A),X6))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(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_4391_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs)) = collect(list(A),aa(list(A),fun(list(A),bool),aTP_Lamp_kd(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).

% set_n_lists
tff(fact_4392_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F2: fun(A,B),I: A] :
          ( pp(aa(set(A),bool,finite_finite(A),F4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_ke(set(A),fun(fun(A,B),fun(A,bool)),I5),F2))),F4))
           => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),I5)),aa(A,B,F2,I)) ) )
              & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),I5) ) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_4393_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_4394_count__notin,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(A,nat,count_list(A,Xs),X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_4395_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P2: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),aa(set(B),set(B),insert(B,I),I5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),I5) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),aa(set(B),set(B),insert(B,I),I5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P2,I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),I5)) ) ) ) ) ) ).

% sum.insert'
tff(fact_4396_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),collect(B,aa(set(B),fun(B,bool),aTP_Lamp_kf(fun(B,A),fun(set(B),fun(B,bool)),G),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),I5) ) ).

% sum.non_neutral'
tff(fact_4397_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),T5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,G,X2) = zero_zero(A) ) )
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                 => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_4398_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),T5: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,H,I3) = zero_zero(A) ) )
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                 => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_4399_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),T5: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,G,X2) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),S3) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_4400_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),T5: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,G,X2) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),T5) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_4401_sum_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),G))))
         => ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_kg(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_4402_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P2: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P2),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),I5) = zero_zero(A) ) ) ) ) ).

% sum.G_def
tff(fact_4403_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),X)),aa(list(A),nat,size_size(list(A)),Xs))) ).

% count_le_length
tff(fact_4404_length__n__lists__elem,axiom,
    ! [A: $tType,Ys2: list(A),N: nat,Xs: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys2),aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs))))
     => ( aa(list(A),nat,size_size(list(A)),Ys2) = N ) ) ).

% length_n_lists_elem
tff(fact_4405_length__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_n_lists
tff(fact_4406_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),I: A] :
          ( pp(aa(set(A),bool,finite_finite(A),collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_ke(set(A),fun(fun(A,B),fun(A,bool)),I5),F2))))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),I5)),aa(A,B,F2,I)) ) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),I5) ) ) ) ) ) ).

% sum_diff1'
tff(fact_4407_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set(nat)] :
      ( ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),M3))
         => ? [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N4))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N4),S3)) ) )
     => ~ pp(aa(set(nat),bool,finite_finite(nat),S3)) ) ).

% unbounded_k_infinite
tff(fact_4408_infinite__nat__iff__unbounded,axiom,
    ! [S3: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite(nat),S3))
    <=> ! [M5: nat] :
        ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N5))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N5),S3)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_4409_integer__of__num_I3_J,axiom,
    ! [N: num] : code_integer_of_num(aa(num,num,bit1,N)) = 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),code_integer_of_num(N)),code_integer_of_num(N))),one_one(code_integer)) ).

% integer_of_num(3)
tff(fact_4410_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),product_Pair(code_integer,bool,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,bool,fNot,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),dvd_dvd(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),K))) ).

% bit_cut_integer_def
tff(fact_4411_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = aa(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),divide_divide(code_integer),K),L)),modulo_modulo(code_integer,K,L)) ).

% divmod_integer_def
tff(fact_4412_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_4413_Re__divide__of__nat,axiom,
    ! [Z: complex,N: nat] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(nat,complex,semiring_1_of_nat(complex),N))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),aa(nat,real,semiring_1_of_nat(real),N)) ).

% Re_divide_of_nat
tff(fact_4414_Re__divide__of__real,axiom,
    ! [Z: complex,R: real] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),real_Vector_of_real(complex,R))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),R) ).

% Re_divide_of_real
tff(fact_4415_Re__sgn,axiom,
    ! [Z: complex] : re(aa(complex,complex,sgn_sgn(complex),Z)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),real_V7770717601297561774m_norm(complex,Z)) ).

% Re_sgn
tff(fact_4416_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(num,complex,numeral_numeral(complex),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),aa(num,real,numeral_numeral(real),W)) ).

% Re_divide_numeral
tff(fact_4417_one__complex_Osimps_I1_J,axiom,
    re(one_one(complex)) = one_one(real) ).

% one_complex.simps(1)
tff(fact_4418_uminus__complex_Osimps_I1_J,axiom,
    ! [X: complex] : re(aa(complex,complex,uminus_uminus(complex),X)) = aa(real,real,uminus_uminus(real),re(X)) ).

% uminus_complex.simps(1)
tff(fact_4419_minus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y2: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y2)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),re(X)),re(Y2)) ).

% minus_complex.simps(1)
tff(fact_4420_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_4421_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),zero_zero(real)))
    <=> ( re(Z) = aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_4422_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),product_Pair(code_integer,bool,zero_zero(code_integer)),fFalse) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( code_bit_cut_integer(K) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,bool),product_case_prod(code_integer,code_integer,product_prod(code_integer,bool),aTP_Lamp_kh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_4423_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),if(real,aa(real,bool,aa(real,fun(real,bool),fequal(real),im(Z)),zero_zero(real)),one_one(real),aa(real,real,sgn_sgn(real),im(Z)))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% csqrt.code
tff(fact_4424_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] : im(csqrt(Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),if(real,aa(real,bool,aa(real,fun(real,bool),fequal(real),im(Z)),zero_zero(real)),one_one(real),aa(real,real,sgn_sgn(real),im(Z)))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% csqrt.simps(2)
tff(fact_4425_Complex__divide,axiom,
    ! [X: complex,Y2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y2) = 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(X)),re(Y2))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y2)),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(X)),re(Y2))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% Complex_divide
tff(fact_4426_Im__divide__of__real,axiom,
    ! [Z: complex,R: real] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),real_Vector_of_real(complex,R))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),R) ).

% Im_divide_of_real
tff(fact_4427_Im__sgn,axiom,
    ! [Z: complex] : im(aa(complex,complex,sgn_sgn(complex),Z)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),real_V7770717601297561774m_norm(complex,Z)) ).

% Im_sgn
tff(fact_4428_Re__i__times,axiom,
    ! [Z: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z)) = aa(real,real,uminus_uminus(real),im(Z)) ).

% Re_i_times
tff(fact_4429_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(num,complex,numeral_numeral(complex),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),aa(num,real,numeral_numeral(real),W)) ).

% Im_divide_numeral
tff(fact_4430_Im__divide__of__nat,axiom,
    ! [Z: complex,N: nat] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(nat,complex,semiring_1_of_nat(complex),N))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),aa(nat,real,semiring_1_of_nat(real),N)) ).

% Im_divide_of_nat
tff(fact_4431_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),im(X)),zero_zero(real)))
        | ( ( im(X) = zero_zero(real) )
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(X))) ) )
     => ( csqrt(aa(complex,complex,uminus_uminus(complex),X)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),csqrt(X)) ) ) ).

% csqrt_minus
tff(fact_4432_imaginary__unit_Osimps_I2_J,axiom,
    im(imaginary_unit) = one_one(real) ).

% imaginary_unit.simps(2)
tff(fact_4433_one__complex_Osimps_I2_J,axiom,
    im(one_one(complex)) = zero_zero(real) ).

% one_complex.simps(2)
tff(fact_4434_uminus__complex_Osimps_I2_J,axiom,
    ! [X: complex] : im(aa(complex,complex,uminus_uminus(complex),X)) = aa(real,real,uminus_uminus(real),im(X)) ).

% uminus_complex.simps(2)
tff(fact_4435_minus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y2: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y2)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),im(X)),im(Y2)) ).

% minus_complex.simps(2)
tff(fact_4436_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y2: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y2)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),re(Y2))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y2))) ).

% times_complex.simps(1)
tff(fact_4437_uminus__complex_Ocode,axiom,
    ! [X: complex] : aa(complex,complex,uminus_uminus(complex),X) = complex2(aa(real,real,uminus_uminus(real),re(X)),aa(real,real,uminus_uminus(real),im(X))) ).

% uminus_complex.code
tff(fact_4438_minus__complex_Ocode,axiom,
    ! [X: complex,Y2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y2) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),re(X)),re(Y2)),aa(real,real,aa(real,fun(real,real),minus_minus(real),im(X)),im(Y2))) ).

% minus_complex.code
tff(fact_4439_times__complex_Ocode,axiom,
    ! [X: complex,Y2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y2) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),re(Y2))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y2))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y2)))) ).

% times_complex.code
tff(fact_4440_Re__power2,axiom,
    ! [X: complex] : re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),X),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(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% Re_power2
tff(fact_4441_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = aa(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),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))),modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))) ).

% divmod_abs_def
tff(fact_4442_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : re(aa(complex,complex,inverse_inverse(complex),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% inverse_complex.simps(1)
tff(fact_4443_Re__divide,axiom,
    ! [X: complex,Y2: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y2)) = 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(X)),re(Y2))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Re_divide
tff(fact_4444_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : im(aa(complex,complex,inverse_inverse(complex),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),im(X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% inverse_complex.simps(2)
tff(fact_4445_Im__divide,axiom,
    ! [X: complex,Y2: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y2)) = 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(X)),re(Y2))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Im_divide
tff(fact_4446_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_4447_inverse__complex_Ocode,axiom,
    ! [X: complex] : aa(complex,complex,inverse_inverse(complex),X) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),re(X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(X)),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(X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% inverse_complex.code
tff(fact_4448_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
               => ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_ki(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( ( L = zero_zero(code_integer) )
               => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),K) ) )
              & ( ( L != zero_zero(code_integer) )
               => ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),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),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_kj(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_4449_Im__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),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_4450_Re__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),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_4451_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_4452_Reals__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,uminus_uminus(A),A3)),real_Vector_Reals(A)))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_Reals(A))) ) ) ).

% Reals_minus_iff
tff(fact_4453_complex__cnj__divide,axiom,
    ! [X: complex,Y2: complex] : cnj(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y2)) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),cnj(X)),cnj(Y2)) ).

% complex_cnj_divide
tff(fact_4454_complex__cnj__one,axiom,
    cnj(one_one(complex)) = one_one(complex) ).

% complex_cnj_one
tff(fact_4455_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( cnj(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% complex_cnj_one_iff
tff(fact_4456_complex__cnj__minus,axiom,
    ! [X: complex] : cnj(aa(complex,complex,uminus_uminus(complex),X)) = aa(complex,complex,uminus_uminus(complex),cnj(X)) ).

% complex_cnj_minus
tff(fact_4457_complex__cnj__diff,axiom,
    ! [X: complex,Y2: complex] : cnj(aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y2)) = aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),cnj(X)),cnj(Y2)) ).

% complex_cnj_diff
tff(fact_4458_complex__cnj__i,axiom,
    cnj(imaginary_unit) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% complex_cnj_i
tff(fact_4459_complex__cnj__neg__numeral,axiom,
    ! [W: num] : cnj(aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W))) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) ).

% complex_cnj_neg_numeral
tff(fact_4460_Re__divide__Reals,axiom,
    ! [R: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R),real_Vector_Reals(complex)))
     => ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),R)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),re(R)) ) ) ).

% Re_divide_Reals
tff(fact_4461_Im__divide__Reals,axiom,
    ! [R: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R),real_Vector_Reals(complex)))
     => ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),R)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),re(R)) ) ) ).

% Im_divide_Reals
tff(fact_4462_Reals__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,semiring_1_of_nat(A),N)),real_Vector_Reals(A))) ) ).

% Reals_of_nat
tff(fact_4463_Reals__minus,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_Reals(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,uminus_uminus(A),A3)),real_Vector_Reals(A))) ) ) ).

% Reals_minus
tff(fact_4464_Reals__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),real_Vector_Reals(A))) ) ).

% Reals_0
tff(fact_4465_Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_Reals(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_Reals(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),real_Vector_Reals(A))) ) ) ) ).

% Reals_divide
tff(fact_4466_cot__in__Reals,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),real_Vector_Reals(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,cot(A),Z)),real_Vector_Reals(A))) ) ) ).

% cot_in_Reals
tff(fact_4467_cos__in__Reals,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),real_Vector_Reals(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,cos(A),Z)),real_Vector_Reals(A))) ) ) ).

% cos_in_Reals
tff(fact_4468_sin__in__Reals,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),real_Vector_Reals(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,sin(A),Z)),real_Vector_Reals(A))) ) ) ).

% sin_in_Reals
tff(fact_4469_fact__in__Reals,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),semiring_char_0_fact(A,N)),real_Vector_Reals(A))) ) ).

% fact_in_Reals
tff(fact_4470_exp__in__Reals,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),real_Vector_Reals(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,exp(A),Z)),real_Vector_Reals(A))) ) ) ).

% exp_in_Reals
tff(fact_4471_tan__in__Reals,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),real_Vector_Reals(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,tan(A),Z)),real_Vector_Reals(A))) ) ) ).

% tan_in_Reals
tff(fact_4472_Reals__1,axiom,
    ! [B: $tType] :
      ( real_V2191834092415804123ebra_1(B)
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),one_one(B)),real_Vector_Reals(B))) ) ).

% Reals_1
tff(fact_4473_Reals__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_Reals(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_Reals(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),real_Vector_Reals(A))) ) ) ) ).

% Reals_diff
tff(fact_4474_cnj_Osimps_I2_J,axiom,
    ! [Z: complex] : im(cnj(Z)) = aa(real,real,uminus_uminus(real),im(Z)) ).

% cnj.simps(2)
tff(fact_4475_complex__cnj,axiom,
    ! [A3: real,B2: real] : cnj(complex2(A3,B2)) = complex2(A3,aa(real,real,uminus_uminus(real),B2)) ).

% complex_cnj
tff(fact_4476_cis__cnj,axiom,
    ! [T2: real] : cnj(cis(T2)) = cis(aa(real,real,uminus_uminus(real),T2)) ).

% cis_cnj
tff(fact_4477_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_Reals(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_Reals(A)))
           => ( ( B2 != zero_zero(A) )
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),real_Vector_Reals(A))) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_4478_nonzero__Reals__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_Reals(A)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,inverse_inverse(A),A3)),real_Vector_Reals(A))) ) ) ) ).

% nonzero_Reals_inverse
tff(fact_4479_cnj_Ocode,axiom,
    ! [Z: complex] : cnj(Z) = complex2(re(Z),aa(real,real,uminus_uminus(real),im(Z))) ).

% cnj.code
tff(fact_4480_Re__complex__div__eq__0,axiom,
    ! [A3: complex,B2: complex] :
      ( ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2)) = zero_zero(real) )
    <=> ( re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))) = zero_zero(real) ) ) ).

% Re_complex_div_eq_0
tff(fact_4481_Im__complex__div__eq__0,axiom,
    ! [A3: complex,B2: complex] :
      ( ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2)) = zero_zero(real) )
    <=> ( im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))) = zero_zero(real) ) ) ).

% Im_complex_div_eq_0
tff(fact_4482_Re__complex__div__lt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),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_4483_Re__complex__div__gt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),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_4484_Re__complex__div__le__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2)))),zero_zero(real))) ) ).

% Re_complex_div_le_0
tff(fact_4485_Re__complex__div__ge__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))))) ) ).

% Re_complex_div_ge_0
tff(fact_4486_Im__complex__div__lt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),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_4487_Im__complex__div__gt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),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_4488_Im__complex__div__le__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2)))),zero_zero(real))) ) ).

% Im_complex_div_le_0
tff(fact_4489_Im__complex__div__ge__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))))) ) ).

% Im_complex_div_ge_0
tff(fact_4490_complex__div__gt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))))) )
      & ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B2))))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),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_4491_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_4492_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,fun(A,A)),V2: num,N: nat] : aa(nat,A,rec_nat(A,A3,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N)) = aa(A,A,aa(nat,fun(A,A),F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N)),aa(nat,A,rec_nat(A,A3,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N))) ).

% rec_nat_add_eq_if
tff(fact_4493_or__int__rec,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fdisj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),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_4494_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero(nat)) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_4495_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y2: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa) = Y2 )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
         => ( Y2 = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa)) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
         => ( Y2 = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_4496_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_4497_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_4498_old_Onat_Osimps_I7_J,axiom,
    ! [T: $tType,F1: T,F22: fun(nat,fun(T,T)),Nat: nat] : aa(nat,T,rec_nat(T,F1,F22),aa(nat,nat,suc,Nat)) = aa(T,T,aa(nat,fun(T,T),F22,Nat),aa(nat,T,rec_nat(T,F1,F22),Nat)) ).

% old.nat.simps(7)
tff(fact_4499_old_Onat_Osimps_I6_J,axiom,
    ! [T: $tType,F1: T,F22: fun(nat,fun(T,T))] : aa(nat,T,rec_nat(T,F1,F22),zero_zero(nat)) = F1 ).

% old.nat.simps(6)
tff(fact_4500_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_4501_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_4502_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_4503_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% or_negative_int_iff
tff(fact_4504_rec__nat__numeral,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,fun(A,A)),V2: num] : aa(nat,A,rec_nat(A,A3,F2),aa(num,nat,numeral_numeral(nat),V2)) = aa(A,A,aa(nat,fun(A,A),F2,pred_numeral(V2)),aa(nat,A,rec_nat(A,A3,F2),pred_numeral(V2))) ).

% rec_nat_numeral
tff(fact_4505_or__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y2: 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,Y2))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y2)) ) ).

% or_numerals(2)
tff(fact_4506_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(8)
tff(fact_4507_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_4508_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_4509_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(5)
tff(fact_4510_or__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y2: 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,Y2))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y2)) ) ).

% or_numerals(1)
tff(fact_4511_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(2)
tff(fact_4512_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(6)
tff(fact_4513_or__minus__minus__numerals,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M2)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_4514_and__minus__minus__numerals,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M2)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_4515_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y2: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y2))) = 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),X)),aa(num,A,numeral_numeral(A),Y2)))) ) ).

% or_numerals(4)
tff(fact_4516_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y2: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y2))) = 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),X)),aa(num,A,numeral_numeral(A),Y2)))) ) ).

% or_numerals(6)
tff(fact_4517_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y2: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y2))) = 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),X)),aa(num,A,numeral_numeral(A),Y2)))) ) ).

% or_numerals(7)
tff(fact_4518_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),zero_zero(A)) = X ) ).

% bit.disj_zero_right
tff(fact_4519_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_4520_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_or_eq
tff(fact_4521_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))) ) ).

% or_greater_eq
tff(fact_4522_OR__lower,axiom,
    ! [X: int,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y2))) ) ) ).

% OR_lower
tff(fact_4523_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_4524_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_4525_set__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se5668285175392031749et_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),aa(int,int,bit_se4730199178511100633sh_bit(int,N),one_one(int))) ).

% set_bit_int_def
tff(fact_4526_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Y2) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Y2) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( X = Y2 ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_4527_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N))) ).

% or_not_numerals(2)
tff(fact_4528_or__not__numerals_I4_J,axiom,
    ! [M2: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M2))),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_4529_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y2) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y2) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y2 ) ) ) ) ).

% bit.compl_unique
tff(fact_4530_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N))) ).

% or_not_numerals(3)
tff(fact_4531_or__not__numerals_I7_J,axiom,
    ! [M2: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M2))),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_4532_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),bit_se2239418461657761734s_mask(A,N)) ) ).

% mask_Suc_exp
tff(fact_4533_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(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))) ) ).

% one_or_eq
tff(fact_4534_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(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))) ) ).

% or_one_eq
tff(fact_4535_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,N))) ) ).

% mask_Suc_double
tff(fact_4536_OR__upper,axiom,
    ! [X: int,N: nat,Y2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ) ).

% OR_upper
tff(fact_4537_or__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(5)
tff(fact_4538_or__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(9)
tff(fact_4539_or__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(8)
tff(fact_4540_prod__decode__aux_Osimps,axiom,
    ! [M2: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),K))
       => ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M2) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),M2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),K))
       => ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M2) = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_4541_old_Orec__nat__def,axiom,
    ! [T: $tType,X3: T,Xa2: fun(nat,fun(T,T)),Xb: nat] : aa(nat,T,rec_nat(T,X3,Xa2),Xb) = the(T,rec_set_nat(T,X3,Xa2,Xb)) ).

% old.rec_nat_def
tff(fact_4542_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y2: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y2 )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y2 = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
        & ( ( Xa != zero_zero(nat) )
         => ( Y2 = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa))))) ) ) ) ) ).

% bezw.elims
tff(fact_4543_bezw_Osimps,axiom,
    ! [Y2: nat,X: nat] :
      ( ( ( Y2 = zero_zero(nat) )
       => ( bezw(X,Y2) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
      & ( ( Y2 != zero_zero(nat) )
       => ( bezw(X,Y2) = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Y2,modulo_modulo(nat,X,Y2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y2,modulo_modulo(nat,X,Y2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y2,modulo_modulo(nat,X,Y2)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y2))))) ) ) ) ).

% bezw.simps
tff(fact_4544_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_4545_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_4546_or__nat__numerals_I2_J,axiom,
    ! [Y2: 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,Y2))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y2)) ).

% or_nat_numerals(2)
tff(fact_4547_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(4)
tff(fact_4548_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_div_numeral
tff(fact_4549_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(3)
tff(fact_4550_or__nat__numerals_I1_J,axiom,
    ! [Y2: 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,Y2))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y2)) ).

% or_nat_numerals(1)
tff(fact_4551_or__minus__numerals_I8_J,axiom,
    ! [N: num,M2: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M2)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M2,aa(num,num,bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_4552_or__minus__numerals_I4_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M2,aa(num,num,bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_4553_or__minus__numerals_I7_J,axiom,
    ! [N: num,M2: 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,N)))),aa(num,int,numeral_numeral(int),M2)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M2,bitM(N)))) ).

% or_minus_numerals(7)
tff(fact_4554_or__minus__numerals_I3_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M2,bitM(N)))) ).

% or_minus_numerals(3)
tff(fact_4555_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_4556_set__bit__nat__def,axiom,
    ! [M2: nat,N: nat] : bit_se5668285175392031749et_bit(nat,M2,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M2),one_one(nat))) ).

% set_bit_nat_def
tff(fact_4557_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M2,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) ) ).

% fst_divmod
tff(fact_4558_or__nat__def,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% or_nat_def
tff(fact_4559_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] : fChoice(product_prod(A,B),product_case_prod(A,B,bool,P)) = fChoice(product_prod(A,B),aTP_Lamp_kk(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P)) ).

% Eps_case_prod
tff(fact_4560_numeral__or__not__num__eq,axiom,
    ! [M2: num,N: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M2,N)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% numeral_or_not_num_eq
tff(fact_4561_int__numeral__not__or__num__neg,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M2))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(N,M2))) ).

% int_numeral_not_or_num_neg
tff(fact_4562_int__numeral__or__not__num__neg,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M2,N))) ).

% int_numeral_or_not_num_neg
tff(fact_4563_floor__real__def,axiom,
    ! [X: real] : archim6421214686448440834_floor(real,X) = the(int,aTP_Lamp_kl(real,fun(int,bool),X)) ).

% floor_real_def
tff(fact_4564_Suc__0__or__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% Suc_0_or_eq
tff(fact_4565_or__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% or_Suc_0_eq
tff(fact_4566_or__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fdisj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2)),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))))),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),M2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% or_nat_rec
tff(fact_4567_bezw__non__0,axiom,
    ! [Y2: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y2))
     => ( bezw(X,Y2) = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Y2,modulo_modulo(nat,X,Y2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y2,modulo_modulo(nat,X,Y2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y2,modulo_modulo(nat,X,Y2)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y2))))) ) ) ).

% bezw_non_0
tff(fact_4568_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_4569_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y2: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y2 )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa))
       => ~ ( ( ( ( Xa = zero_zero(nat) )
               => ( Y2 = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
              & ( ( Xa != zero_zero(nat) )
               => ( Y2 = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa))))) ) ) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa)) ) ) ) ).

% bezw.pelims
tff(fact_4570_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
          | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) )
      & ( ~ ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
            | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( ( ( K = zero_zero(int) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = L ) )
          & ( ( K != zero_zero(int) )
           => ( ( ( L = zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = K ) )
              & ( ( L != zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,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_4571_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_4572_max_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).

% max.right_idem
tff(fact_4573_max_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).

% max.left_idem
tff(fact_4574_max_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),A3) = A3 ) ).

% max.idem
tff(fact_4575_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).

% max.bounded_iff
tff(fact_4576_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_4577_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb1
tff(fact_4578_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb3
tff(fact_4579_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_4580_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),Z)) ) ) ) ).

% max_less_iff_conj
tff(fact_4581_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fdisj(P,Q)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_or_iff
tff(fact_4582_fst__divmod__nat,axiom,
    ! [M2: nat,N: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(M2,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) ).

% fst_divmod_nat
tff(fact_4583_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_integer(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),L) ).

% fst_divmod_integer
tff(fact_4584_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(1)
tff(fact_4585_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(3)
tff(fact_4586_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(4)
tff(fact_4587_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_4588_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_4589_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(5)
tff(fact_4590_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(6)
tff(fact_4591_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( 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),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( 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),V2))) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(2)
tff(fact_4592_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( 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),V2)) = aa(num,A,numeral_numeral(A),V2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( 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),V2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(3)
tff(fact_4593_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( 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),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( 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),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(4)
tff(fact_4594_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_abs(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L)) ).

% fst_divmod_abs
tff(fact_4595_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_4596_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Xa2) = Xa2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Xa2) = X3 ) ) ) ) ).

% max_def_raw
tff(fact_4597_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y2: 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),X),Y2)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y2),Z)) ) ).

% max_diff_distrib_left
tff(fact_4598_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,D2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_4599_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.orderE
tff(fact_4600_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) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% max.orderI
tff(fact_4601_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).

% max.boundedE
tff(fact_4602_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)) ) ) ) ).

% max.boundedI
tff(fact_4603_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4604_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).

% max.cobounded1
tff(fact_4605_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).

% max.cobounded2
tff(fact_4606_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y2)) ) ) ) ).

% le_max_iff_disj
tff(fact_4607_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4608_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4609_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.coboundedI1
tff(fact_4610_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.coboundedI2
tff(fact_4611_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_max(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% max_add_distrib_right
tff(fact_4612_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y2: 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),X),Y2)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y2),Z)) ) ).

% max_add_distrib_left
tff(fact_4613_of__int__max,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Y2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(int,A,ring_1_of_int(A),X)),aa(int,A,ring_1_of_int(A),Y2)) ) ).

% of_int_max
tff(fact_4614_max_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).

% max.left_commute
tff(fact_4615_max_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3) ) ).

% max.commute
tff(fact_4616_max_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).

% max.assoc
tff(fact_4617_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),Y2)) ) ) ) ).

% less_max_iff_disj
tff(fact_4618_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% max.strict_boundedE
tff(fact_4619_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4620_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.strict_coboundedI1
tff(fact_4621_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.strict_coboundedI2
tff(fact_4622_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y2: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y2)) ) ).

% of_nat_max
tff(fact_4623_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y2: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa) = Y2 )
     => ( accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
               => ( Y2 = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa)) ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
               => ( Y2 = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,suc,X))) ) ) )
           => ~ accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_4624_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_km(rat,fun(int,bool),X)) ).

% floor_rat_def
tff(fact_4625_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),N: nat,Xs: list(A)] :
      ( pp(aa(set(product_prod(nat,A)),bool,aa(product_prod(nat,A),fun(set(product_prod(nat,A)),bool),member(product_prod(nat,A)),P2),aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N,Xs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),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)),N)))
        & ( 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)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_4626_max__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M2),N)) ).

% max_Suc_Suc
tff(fact_4627_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_4628_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_4629_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_4630_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_4631_max__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N) = N ).

% max_0L
tff(fact_4632_max__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),zero_zero(nat)) = N ).

% max_0R
tff(fact_4633_length__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,N,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_enumerate
tff(fact_4634_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),N)) ).

% max_numeral_Suc
tff(fact_4635_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_4636_nat__mult__max__left,axiom,
    ! [M2: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M2),N)),Q5) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)) ).

% nat_mult_max_left
tff(fact_4637_nat__mult__max__right,axiom,
    ! [M2: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q5)) ).

% nat_mult_max_right
tff(fact_4638_nat__add__max__right,axiom,
    ! [M2: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),Q5)) ).

% nat_add_max_right
tff(fact_4639_nat__add__max__left,axiom,
    ! [M2: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M2),N)),Q5) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),Q5)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q5)) ).

% nat_add_max_left
tff(fact_4640_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R))
     => ~ ! [S2: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),S2))
           => ! [T3: rat] :
                ( pp(aa(rat,bool,aa(rat,fun(rat,bool),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_4641_sgn__rat__def,axiom,
    ! [A3: rat] :
      ( ( ( A3 = zero_zero(rat) )
       => ( aa(rat,rat,sgn_sgn(rat),A3) = zero_zero(rat) ) )
      & ( ( A3 != zero_zero(rat) )
       => ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
           => ( aa(rat,rat,sgn_sgn(rat),A3) = one_one(rat) ) )
          & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
           => ( aa(rat,rat,sgn_sgn(rat),A3) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_4642_abs__rat__def,axiom,
    ! [A3: rat] :
      ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A3),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A3) = aa(rat,rat,uminus_uminus(rat),A3) ) )
      & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A3),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A3) = A3 ) ) ) ).

% abs_rat_def
tff(fact_4643_nat__minus__add__max,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)),M2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),M2) ).

% nat_minus_add_max
tff(fact_4644_max__Suc1,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),M2) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_kn(nat,fun(nat,nat),N),M2) ).

% max_Suc1
tff(fact_4645_max__Suc2,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M2),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_ko(nat,fun(nat,nat),N),M2) ).

% max_Suc2
tff(fact_4646_nth__enumerate__eq,axiom,
    ! [A: $tType,M2: nat,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N,Xs)),M2) = aa(A,product_prod(nat,A),product_Pair(nat,A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),aa(nat,A,nth(A,Xs),M2)) ) ) ).

% nth_enumerate_eq
tff(fact_4647_or__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N) = N ) )
      & ( ( M2 != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N) = M2 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N) = 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,M2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,N,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),M2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_4648_rec__nat__0__imp,axiom,
    ! [A: $tType,F2: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F2 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F2,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_4649_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F2: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),N: nat] :
      ( ( F2 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),F22,N),aa(nat,A,F2,N)) ) ) ).

% rec_nat_Suc_imp
tff(fact_4650_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_kp(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_4651_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(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_4652_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_4653_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_4654_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),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_4655_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).

% quotient_of_number(4)
tff(fact_4656_diff__rat__def,axiom,
    ! [Q5: rat,R: rat] : aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q5),R) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q5),aa(rat,rat,uminus_uminus(rat),R)) ).

% diff_rat_def
tff(fact_4657_divide__rat__def,axiom,
    ! [Q5: rat,R: rat] : aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),Q5),R) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Q5),aa(rat,rat,inverse_inverse(rat),R)) ).

% divide_rat_def
tff(fact_4658_quotient__of__div,axiom,
    ! [R: rat,N: int,D2: int] :
      ( ( quotient_of(R) = aa(int,product_prod(int,int),product_Pair(int,int,N),D2) )
     => ( R = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,ring_1_of_int(rat),N)),aa(int,rat,ring_1_of_int(rat),D2)) ) ) ).

% quotient_of_div
tff(fact_4659_quotient__of__denom__pos,axiom,
    ! [R: rat,P2: int,Q5: int] :
      ( ( quotient_of(R) = aa(int,product_prod(int,int),product_Pair(int,int,P2),Q5) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q5)) ) ).

% quotient_of_denom_pos
tff(fact_4660_quotient__of__denom__pos_H,axiom,
    ! [R: rat] : pp(aa(int,bool,aa(int,fun(int,bool),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_4661_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_kq(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_4662_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P2)) = aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P2)))),one_one(int)) ).

% rat_sgn_code
tff(fact_4663_rat__less__code,axiom,
    ! [P2: rat,Q5: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),P2),Q5))
    <=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aTP_Lamp_ks(rat,fun(int,fun(int,bool)),Q5)),quotient_of(P2))) ) ).

% rat_less_code
tff(fact_4664_quotient__of__int,axiom,
    ! [A3: int] : quotient_of(of_int(A3)) = aa(int,product_prod(int,int),product_Pair(int,int,A3),one_one(int)) ).

% quotient_of_int
tff(fact_4665_rat__minus__code,axiom,
    ! [P2: rat,Q5: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P2),Q5)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ku(rat,fun(int,fun(int,product_prod(int,int))),Q5)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_4666_normalize__negative,axiom,
    ! [Q5: int,P2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Q5),zero_zero(int)))
     => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P2),Q5)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q5))) ) ) ).

% normalize_negative
tff(fact_4667_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa(int,product_prod(int,int),product_Pair(int,int,P2),zero_zero(int))) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_4668_normalize__denom__pos,axiom,
    ! [R: product_prod(int,int),P2: int,Q5: int] :
      ( ( normalize(R) = aa(int,product_prod(int,int),product_Pair(int,int,P2),Q5) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q5)) ) ).

% normalize_denom_pos
tff(fact_4669_normalize__crossproduct,axiom,
    ! [Q5: int,S: int,P2: int,R: int] :
      ( ( Q5 != zero_zero(int) )
     => ( ( S != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P2),Q5)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,R),S)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S) = aa(int,int,aa(int,fun(int,int),times_times(int),R),Q5) ) ) ) ) ).

% normalize_crossproduct
tff(fact_4670_rat__divide__code,axiom,
    ! [P2: rat,Q5: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P2),Q5)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_kw(rat,fun(int,fun(int,product_prod(int,int))),Q5)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_4671_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(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_4672_normalize__def,axiom,
    ! [P2: product_prod(int,int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)))
       => ( normalize(P2) = aa(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)),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,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),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)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)))
       => ( ( ( aa(product_prod(int,int),int,product_snd(int,int),P2) = zero_zero(int) )
           => ( normalize(P2) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ) )
          & ( ( aa(product_prod(int,int),int,product_snd(int,int),P2) != zero_zero(int) )
           => ( normalize(P2) = aa(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)),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,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),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))))) ) ) ) ) ) ).

% normalize_def
tff(fact_4673_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(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_4674_gcd__eq__0__iff,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) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_4675_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_4676_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_4677_gcd__neg1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd_neg1
tff(fact_4678_gcd__neg2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd_neg2
tff(fact_4679_gcd__1__int,axiom,
    ! [M2: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M2),one_one(int)) = one_one(int) ).

% gcd_1_int
tff(fact_4680_gcd__neg2__int,axiom,
    ! [X: int,Y2: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y2)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y2) ).

% gcd_neg2_int
tff(fact_4681_gcd__neg1__int,axiom,
    ! [X: int,Y2: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y2) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y2) ).

% gcd_neg1_int
tff(fact_4682_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,N: 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),N))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(num,A,numeral_numeral(A),N)) ) ).

% gcd_neg_numeral_2
tff(fact_4683_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: 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),N))),A3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(num,A,numeral_numeral(A),N)),A3) ) ).

% gcd_neg_numeral_1
tff(fact_4684_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4685_gcd__pos__int,axiom,
    ! [M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M2),N)))
    <=> ( ( M2 != zero_zero(int) )
        | ( N != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_4686_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(num,int,numeral_numeral(int),N)) ).

% gcd_neg_numeral_2_int
tff(fact_4687_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),X) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(num,int,numeral_numeral(int),N)),X) ).

% gcd_neg_numeral_1_int
tff(fact_4688_gcd__0__int,axiom,
    ! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),zero_zero(int)) = aa(int,int,abs_abs(int),X) ).

% gcd_0_int
tff(fact_4689_gcd__0__left__int,axiom,
    ! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),zero_zero(int)),X) = aa(int,int,abs_abs(int),X) ).

% gcd_0_left_int
tff(fact_4690_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: A,M2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),M2)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),N) ) ).

% gcd_diff2
tff(fact_4691_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [M2: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),M2),N)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),N) ) ).

% gcd_diff1
tff(fact_4692_gcd__ge__0__int,axiom,
    ! [X: int,Y2: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y2))) ).

% gcd_ge_0_int
tff(fact_4693_Frct__code__post_I9_J,axiom,
    ! [Q5: product_prod(int,int)] : aa(rat,rat,uminus_uminus(rat),aa(rat,rat,uminus_uminus(rat),frct(Q5))) = frct(Q5) ).

% Frct_code_post(9)
tff(fact_4694_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4695_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4696_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4697_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4698_gcd__le1__int,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),A3)) ) ).

% gcd_le1_int
tff(fact_4699_gcd__le2__int,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),B2)) ) ).

% gcd_le2_int
tff(fact_4700_gcd__cases__int,axiom,
    ! [X: int,Y2: int,P: fun(int,bool)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y2))) ) )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y2),zero_zero(int)))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y2)))) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y2))) ) )
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y2),zero_zero(int)))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y2)))) ) )
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y2))) ) ) ) ) ).

% gcd_cases_int
tff(fact_4701_gcd__unique__int,axiom,
    ! [D2: int,A3: int,B2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D2))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),A3))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),B2))
        & ! [E3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E3),A3))
              & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E3),B2)) )
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E3),D2)) ) )
    <=> ( D2 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2) ) ) ).

% gcd_unique_int
tff(fact_4702_gcd__non__0__int,axiom,
    ! [Y2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Y2))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y2) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y2),modulo_modulo(int,X,Y2)) ) ) ).

% gcd_non_0_int
tff(fact_4703_gcd__code__int,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),L) = aa(int,int,abs_abs(int),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),L),zero_zero(int)),K,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),L),modulo_modulo(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L))))) ).

% gcd_code_int
tff(fact_4704_Frct__code__post_I1_J,axiom,
    ! [A3: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),A3)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_4705_Frct__code__post_I2_J,axiom,
    ! [A3: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,A3),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_4706_Frct__code__post_I8_J,axiom,
    ! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,A3),aa(int,int,uminus_uminus(int),B2))) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),product_Pair(int,int,A3),B2))) ).

% Frct_code_post(8)
tff(fact_4707_Frct__code__post_I7_J,axiom,
    ! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),A3)),B2)) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),product_Pair(int,int,A3),B2))) ).

% Frct_code_post(7)
tff(fact_4708_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_4709_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(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_4710_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_4711_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( ( L = zero_zero(code_integer) )
           => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),K) ) )
          & ( ( L != zero_zero(code_integer) )
           => ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),aa(code_integer,code_integer,sgn_sgn(code_integer),K)),aa(code_integer,code_integer,sgn_sgn(code_integer),L)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_kx(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_4712_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) = A3 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),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_4713_gcd__0__left__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),X) = X ).

% gcd_0_left_nat
tff(fact_4714_gcd__0__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),zero_zero(nat)) = X ).

% gcd_0_nat
tff(fact_4715_gcd__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),zero_zero(nat)) = A3 ).

% gcd_nat.right_neutral
tff(fact_4716_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_4717_gcd__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A3) = A3 ).

% gcd_nat.left_neutral
tff(fact_4718_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_4719_gcd__1__nat,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_4720_gcd__Suc__0,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_4721_gcd__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N)))
    <=> ( ( M2 != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_4722_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_4723_gcd__int__int__eq,axiom,
    ! [M2: nat,N: nat] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N)) ).

% gcd_int_int_eq
tff(fact_4724_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,B2: bool] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(bool,A,zero_neq_one_of_bool(A),B2)) = aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat)),B2)) ) ).

% drop_bit_of_bool
tff(fact_4725_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ).

% drop_bit_of_Suc_0
tff(fact_4726_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4197421643247451524op_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_4727_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se4197421643247451524op_bit(int,N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% drop_bit_negative_int_iff
tff(fact_4728_drop__bit__minus__one,axiom,
    ! [N: nat] : aa(int,int,bit_se4197421643247451524op_bit(int,N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_4729_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_4730_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_4731_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% drop_bit_of_1
tff(fact_4732_gcd__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ).

% gcd_nat_abs_right_eq
tff(fact_4733_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ).

% gcd_nat_abs_left_eq
tff(fact_4734_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_4735_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_4736_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_4737_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M2: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M2)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),M2)) ) ).

% drop_bit_of_nat
tff(fact_4738_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,M2),N)) = aa(A,A,bit_se4197421643247451524op_bit(A,M2),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_drop_bit
tff(fact_4739_gcd__le1__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),A3)) ) ).

% gcd_le1_nat
tff(fact_4740_gcd__le2__nat,axiom,
    ! [B2: nat,A3: nat] :
      ( ( B2 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),B2)) ) ).

% gcd_le2_nat
tff(fact_4741_gcd__diff1__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N) ) ) ).

% gcd_diff1_nat
tff(fact_4742_gcd__diff2__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N) ) ) ).

% gcd_diff2_nat
tff(fact_4743_gcd__non__0__nat,axiom,
    ! [Y2: nat,X: nat] :
      ( ( Y2 != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y2) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y2),modulo_modulo(nat,X,Y2)) ) ) ).

% gcd_non_0_nat
tff(fact_4744_gcd__nat_Osimps,axiom,
    ! [Y2: nat,X: nat] :
      ( ( ( Y2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y2) = X ) )
      & ( ( Y2 != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y2) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y2),modulo_modulo(nat,X,Y2)) ) ) ) ).

% gcd_nat.simps
tff(fact_4745_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa: nat,Y2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = Y2 )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y2 = X ) )
        & ( ( Xa != zero_zero(nat) )
         => ( Y2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_4746_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( bit_se2584673776208193580ke_bit(A,N,A3) = A3 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_4747_drop__bit__push__bit__int,axiom,
    ! [M2: nat,N: nat,K: int] : aa(int,int,bit_se4197421643247451524op_bit(int,M2),aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)) = aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),aa(int,int,bit_se4730199178511100633sh_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)),K)) ).

% drop_bit_push_bit_int
tff(fact_4748_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,M2),bit_se2584673776208193580ke_bit(A,N,A3)) = bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2),aa(A,A,bit_se4197421643247451524op_bit(A,M2),A3)) ) ).

% drop_bit_take_bit
tff(fact_4749_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) )
         => ( ! [X2: B,Y3: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X2)),aa(B,A,H,Y3))
           => ( aa(set(C),A,aa(fun(C,A),fun(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,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A4)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_4750_bezout__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [X2: nat,Y3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)) ) ).

% bezout_nat
tff(fact_4751_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A3: nat] :
    ? [X2: nat,Y3: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) )
      | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_4752_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_4753_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A3)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_4754_gcd__int__def,axiom,
    ! [X: int,Y2: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y2) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),X))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y2)))) ).

% gcd_int_def
tff(fact_4755_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: 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,N),A3) = A3 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_4756_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),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,N),A3)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% drop_bit_half
tff(fact_4757_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se4197421643247451524op_bit(int,N),K) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% drop_bit_int_def
tff(fact_4758_drop__bit__nat__def,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),M2) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% drop_bit_nat_def
tff(fact_4759_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,bit_se4197421643247451524op_bit(A,N),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_4760_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),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))),N)) ) ).

% drop_bit_eq_div
tff(fact_4761_bezw__aux,axiom,
    ! [X: nat,Y2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(X,Y2))),aa(nat,int,semiring_1_of_nat(int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(X,Y2))),aa(nat,int,semiring_1_of_nat(int),Y2))) ).

% bezw_aux
tff(fact_4762_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = Y2 )
     => ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa))
       => ~ ( ( ( ( Xa = zero_zero(nat) )
               => ( Y2 = X ) )
              & ( ( Xa != zero_zero(nat) )
               => ( Y2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa)) ) ) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_4763_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: 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),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),neg_numeral_sub(int,N,one2))) ).

% xor_minus_numerals(2)
tff(fact_4764_xor__minus__numerals_I1_J,axiom,
    ! [N: 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),N))),K) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),neg_numeral_sub(int,N,one2)),K)) ).

% xor_minus_numerals(1)
tff(fact_4765_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_4766_diff__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,M2,N) ) ).

% diff_numeral_simps(1)
tff(fact_4767_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_4768_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_4769_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,M2) ) ).

% add_neg_numeral_simps(2)
tff(fact_4770_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,M2,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_4771_semiring__norm_I166_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W: num,Y2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V2,W)),Y2) ) ).

% semiring_norm(166)
tff(fact_4772_semiring__norm_I167_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W: num,Y2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Y2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W,V2)),Y2) ) ).

% semiring_norm(167)
tff(fact_4773_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,M2) ) ).

% diff_numeral_simps(4)
tff(fact_4774_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_4775_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_4776_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M2)),one_one(A)) = neg_numeral_sub(A,M2,one2) ) ).

% diff_numeral_special(2)
tff(fact_4777_diff__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,one2,N) ) ).

% diff_numeral_special(1)
tff(fact_4778_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_4779_not__minus__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,one2) ) ).

% not_minus_numeral_eq
tff(fact_4780_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_4781_add__neg__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,one2) ) ).

% add_neg_numeral_special(4)
tff(fact_4782_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M2,one2) ) ).

% add_neg_numeral_special(3)
tff(fact_4783_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A)) = neg_numeral_sub(A,one2,M2) ) ).

% add_neg_numeral_special(2)
tff(fact_4784_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: 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),M2))) = neg_numeral_sub(A,one2,M2) ) ).

% add_neg_numeral_special(1)
tff(fact_4785_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M2,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)) ) ).

% minus_sub_one_diff_one
tff(fact_4786_diff__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: 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),N))) = neg_numeral_sub(A,N,one2) ) ).

% diff_numeral_special(7)
tff(fact_4787_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M2) ) ).

% diff_numeral_special(8)
tff(fact_4788_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_4789_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_4790_card_Ocomp__fun__commute__on,axiom,
    aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) = aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) ).

% card.comp_fun_commute_on
tff(fact_4791_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_4792_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_4793_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_4794_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = 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,M2,N)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_4795_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = 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,M2,N)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_4796_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_ky(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_4797_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,N,M2)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N)) ) ) ).

% sub_non_negative
tff(fact_4798_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),neg_numeral_sub(A,N,M2)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M2)) ) ) ).

% sub_non_positive
tff(fact_4799_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),neg_numeral_sub(A,N,M2)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M2)) ) ) ).

% sub_negative
tff(fact_4800_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),neg_numeral_sub(A,N,M2)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N)) ) ) ).

% sub_positive
tff(fact_4801_sub__inc__One__eq,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : neg_numeral_sub(A,inc(N),one2) = aa(num,A,numeral_numeral(A),N) ) ).

% sub_inc_One_eq
tff(fact_4802_minus__numeral__eq__not__sub__one,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),neg_numeral_sub(A,N,one2)) ) ).

% minus_numeral_eq_not_sub_one
tff(fact_4803_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(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),N))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_4804_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(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),N))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_4805_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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),N))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_4806_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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),N))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_4807_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N)) = aa(set(nat),A,aa(fun(nat,A),fun(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),M2))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_4808_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N)) = 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),M2))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_4809_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kz(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_4810_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kz(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_4811_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kz(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_4812_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kz(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_4813_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M2: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_4814_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M2: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_4815_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M2: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_4816_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = aa(set(nat),A,aa(fun(nat,A),fun(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),M2))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_4817_sub__BitM__One__eq,axiom,
    ! [N: num] : neg_numeral_sub(int,bitM(N),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,N,one2)) ).

% sub_BitM_One_eq
tff(fact_4818_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = 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),M2))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_4819_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M2: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_4820_Code__Numeral_Onegative__def,axiom,
    code_negative = aa(fun(num,code_integer),fun(num,code_integer),comp(code_integer,code_integer,num,uminus_uminus(code_integer)),numeral_numeral(code_integer)) ).

% Code_Numeral.negative_def
tff(fact_4821_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_4822_div__add__self1__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A3: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),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_no_field
tff(fact_4823_div__add__self2__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A3: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_no_field
tff(fact_4824_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y2: A,A3: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2),A3)
         => ( real_V8093663219630862766scaleR(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A3,X)),real_V8093663219630862766scaleR(A,A3,Y2)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_4825_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y2: A,A3: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2),A3)
         => ( real_V8093663219630862766scaleR(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,A3,X)),real_V8093663219630862766scaleR(A,A3,Y2)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_4826_distrib__right__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y2: B,C2: A,A3: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y2),C2)
         => ( 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_NO_MATCH
tff(fact_4827_distrib__left__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y2: B,A3: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y2),A3)
         => ( 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_NO_MATCH
tff(fact_4828_left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y2: B,C2: A,A3: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y2),C2)
         => ( 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_NO_MATCH
tff(fact_4829_right__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y2: B,A3: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y2),A3)
         => ( 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_NO_MATCH
tff(fact_4830_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,N: nat] :
          ( nO_MATCH(A,A,one_one(A),X)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ) ).

% power_minus'
tff(fact_4831_scale__left__distrib__NO__MATCH,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y2: A,C2: C,A3: real,B2: real] :
          ( nO_MATCH(A,C,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2),C2)
         => ( real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),B2),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A3,X)),real_V8093663219630862766scaleR(A,B2,X)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_4832_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y2: A,C2: C,A3: real,B2: real] :
          ( nO_MATCH(A,C,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2),C2)
         => ( real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),B2),X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,A3,X)),real_V8093663219630862766scaleR(A,B2,X)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_4833_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,aa(fun(nat,A),fun(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_4834_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
     => ( code_nat_of_integer(K) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_4835_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jv(fun(nat,A),fun(nat,fun(A,bool)),C2),N))))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jv(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))),N)) ) ) ) ) ).

% polyfun_rootbound
tff(fact_4836_card__Collect__less__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N))) = N ).

% card_Collect_less_nat
tff(fact_4837_Suc__funpow,axiom,
    ! [N: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),N),suc) = aa(nat,fun(nat,nat),plus_plus(nat),N) ).

% Suc_funpow
tff(fact_4838_card__atMost,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),set_ord_atMost(nat,U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_4839_funpow__0,axiom,
    ! [A: $tType,F2: fun(A,A),X: 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),X) = X ).

% funpow_0
tff(fact_4840_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or7035219750837199246ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_atLeastLessThan
tff(fact_4841_card__Collect__le__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_lb(nat,fun(nat,bool)),N))) = aa(nat,nat,suc,N) ).

% card_Collect_le_nat
tff(fact_4842_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_4843_card_Oinfinite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
     => ( aa(set(A),nat,finite_card(A),A4) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_4844_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,U)),L) ).

% card_atLeastAtMost
tff(fact_4845_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_atLeastLessThan_int
tff(fact_4846_card__0__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( ( aa(set(A),nat,finite_card(A),A4) = zero_zero(nat) )
      <=> ( A4 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_4847_card__insert__disjoint,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A4)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_4848_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y2: A,A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_lc(A,fun(B,A),Y2)),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))),Y2) ) ).

% sum_constant
tff(fact_4849_card__Diff__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),B5))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,A3),B5))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_4850_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or1337092689740270186AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_4851_comp__funpow,axiom,
    ! [B: $tType,A: $tType,N: nat,F2: fun(A,A)] : aa(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A)),aa(nat,fun(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A))),compow(fun(fun(B,A),fun(B,A))),N),comp(A,A,B,F2)) = comp(A,A,B,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)) ).

% comp_funpow
tff(fact_4852_funpow__mult,axiom,
    ! [A: $tType,N: nat,M2: 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)),N),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M2),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),M2),N)),F2) ).

% funpow_mult
tff(fact_4853_funpow__swap1,axiom,
    ! [A: $tType,F2: fun(A,A),N: nat,X: 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)),N),F2),X)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),aa(A,A,F2,X)) ).

% funpow_swap1
tff(fact_4854_bij__betw__funpow,axiom,
    ! [A: $tType,F2: fun(A,A),S3: set(A),N: 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)),N),F2),S3,S3) ) ).

% bij_betw_funpow
tff(fact_4855_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,N: 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,N)),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)),N),F2)) ).

% funpow.simps(2)
tff(fact_4856_funpow__Suc__right,axiom,
    ! [A: $tType,N: 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,N)),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)),N),F2)),F2) ).

% funpow_Suc_right
tff(fact_4857_funpow__add,axiom,
    ! [A: $tType,M2: nat,N: 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),M2),N)),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)),M2),F2)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)) ).

% funpow_add
tff(fact_4858_card__lists__length__eq,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_iy(set(A),fun(nat,fun(list(A),bool)),A4),N))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_4859_card__eq__sum,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),nat,finite_card(A),A4) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ld(A,nat)),A4) ).

% card_eq_sum
tff(fact_4860_card__eq__0__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = zero_zero(nat) )
    <=> ( ( A4 = bot_bot(set(A)) )
        | ~ pp(aa(set(A),bool,finite_finite(A),A4)) ) ) ).

% card_eq_0_iff
tff(fact_4861_card__ge__0__finite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4)))
     => pp(aa(set(A),bool,finite_finite(A),A4)) ) ).

% card_ge_0_finite
tff(fact_4862_card__insert__if,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A4)) = aa(set(A),nat,finite_card(A),A4) ) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A4)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A4)) ) ) ) ) ).

% card_insert_if
tff(fact_4863_card__Suc__eq__finite,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,K) )
    <=> ? [B3: A,B8: set(A)] :
          ( ( A4 = aa(set(A),set(A),insert(A,B3),B8) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),B8))
          & ( aa(set(A),nat,finite_card(A),B8) = K )
          & pp(aa(set(A),bool,finite_finite(A),B8)) ) ) ).

% card_Suc_eq_finite
tff(fact_4864_card__less__sym__Diff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,finite_finite(A),B5))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B5)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),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),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4865_card__le__sym__Diff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,finite_finite(A),B5))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B5)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),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),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4866_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_4867_card__1__singletonE,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = one_one(nat) )
     => ~ ! [X2: A] : A4 != aa(set(A),set(A),insert(A,X2),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_4868_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M10: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),M10))
     => ? [H2: fun(A,nat)] : bij_betw(A,nat,H2,M10,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M10))) ) ).

% ex_bij_betw_finite_nat
tff(fact_4869_psubset__card__mono,axiom,
    ! [A: $tType,B5: set(A),A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B5))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B5))) ) ) ).

% psubset_card_mono
tff(fact_4870_card__less__Suc2,axiom,
    ! [M10: set(nat),I: nat] :
      ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M10))
     => ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_le(set(nat),fun(nat,fun(nat,bool)),M10),I))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_lf(set(nat),fun(nat,fun(nat,bool)),M10),I))) ) ) ).

% card_less_Suc2
tff(fact_4871_card__less__Suc,axiom,
    ! [M10: set(nat),I: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M10))
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_le(set(nat),fun(nat,fun(nat,bool)),M10),I)))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_lf(set(nat),fun(nat,fun(nat,bool)),M10),I))) ) ) ).

% card_less_Suc
tff(fact_4872_card__less,axiom,
    ! [M10: set(nat),I: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M10))
     => ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_lf(set(nat),fun(nat,fun(nat,bool)),M10),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_4873_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) = aa(int,nat,nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_4874_sum__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),A4: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_lg(fun(A,nat),fun(A,nat),F2)),A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(set(A),nat,finite_card(A),A4)) ).

% sum_Suc
tff(fact_4875_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_zero(code_integer)) = zero_zero(nat) ).

% nat_of_integer_code_post(1)
tff(fact_4876_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,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_lh(A,real)),A4) ).

% real_of_card
tff(fact_4877_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),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),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_4878_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_4879_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_4880_sum__bounded__below,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A4: set(B),K6: A,F2: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K6),aa(B,A,F2,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A4))),K6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4))) ) ) ).

% sum_bounded_below
tff(fact_4881_sum__bounded__above,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A4: set(B),F2: fun(B,A),K6: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),K6)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),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))),K6))) ) ) ).

% sum_bounded_above
tff(fact_4882_card__gt__0__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4)))
    <=> ( ( A4 != bot_bot(set(A)) )
        & pp(aa(set(A),bool,finite_finite(A),A4)) ) ) ).

% card_gt_0_iff
tff(fact_4883_card__1__singleton__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X4: A] : A4 = aa(set(A),set(A),insert(A,X4),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_4884_card__eq__SucD,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,K) )
     => ? [B4: A,B7: set(A)] :
          ( ( A4 = aa(set(A),set(A),insert(A,B4),B7) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B7))
          & ( aa(set(A),nat,finite_card(A),B7) = K )
          & ( ( K = zero_zero(nat) )
           => ( B7 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_4885_card__Suc__eq,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,K) )
    <=> ? [B3: A,B8: set(A)] :
          ( ( A4 = aa(set(A),set(A),insert(A,B3),B8) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),B8))
          & ( aa(set(A),nat,finite_card(A),B8) = K )
          & ( ( K = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_4886_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(nat,nat,suc,zero_zero(nat))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
           => ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A4))
               => ( X4 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4887_card__le__Suc__iff,axiom,
    ! [A: $tType,N: nat,A4: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),A4)))
    <=> ? [A5: A,B8: set(A)] :
          ( ( A4 = aa(set(A),set(A),insert(A,A5),B8) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),B8))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),B8)))
          & pp(aa(set(A),bool,finite_finite(A),B8)) ) ) ).

% card_le_Suc_iff
tff(fact_4888_card__Diff1__le,axiom,
    ! [A: $tType,A4: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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,X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4))) ).

% card_Diff1_le
tff(fact_4889_card__Diff__subset,axiom,
    ! [A: $tType,B5: set(A),A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),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),B5)) = 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),B5)) ) ) ) ).

% card_Diff_subset
tff(fact_4890_card__psubset,axiom,
    ! [A: $tType,B5: set(A),A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B5)))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B5)) ) ) ) ).

% card_psubset
tff(fact_4891_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B5: set(A),A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),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),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)))) ) ).

% diff_card_le_card_Diff
tff(fact_4892_card__lists__length__le,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_iz(set(A),fun(nat,fun(list(A),bool)),A4),N))) = aa(set(nat),nat,aa(fun(nat,nat),fun(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,N)) ) ) ).

% card_lists_length_le
tff(fact_4893_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M10: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),M10))
     => ? [H2: fun(nat,A)] : bij_betw(nat,A,H2,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M10)),M10) ) ).

% ex_bij_betw_nat_finite
tff(fact_4894_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M10: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),M10))
     => ? [H2: fun(nat,A)] : bij_betw(nat,A,H2,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M10)),M10) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_4895_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aTP_Lamp_jn(nat,fun(A,bool),N)))),N)) ) ) ).

% card_roots_unity
tff(fact_4896_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N6: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N6)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_4897_card__sum__le__nat__sum,axiom,
    ! [S3: set(nat)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ai(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S3)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ai(nat,nat)),S3))) ).

% card_sum_le_nat_sum
tff(fact_4898_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_4899_card__nth__roots,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( aa(set(complex),nat,finite_card(complex),collect(complex,aa(nat,fun(complex,bool),aTP_Lamp_bi(complex,fun(nat,fun(complex,bool)),C2),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_4900_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(complex),nat,finite_card(complex),collect(complex,aTP_Lamp_bh(nat,fun(complex,bool),N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_4901_card__Suc__Diff1,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4902_card_Oinsert__remove,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A4)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_4903_card_Oremove,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
       => ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_4904_card__Diff1__less,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),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,X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4))) ) ) ).

% card_Diff1_less
tff(fact_4905_card__Diff2__less,axiom,
    ! [A: $tType,A4: set(A),X: A,Y2: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),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,X),bot_bot(set(A))))),aa(set(A),set(A),insert(A,Y2),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4))) ) ) ) ).

% card_Diff2_less
tff(fact_4906_card__Diff1__less__iff,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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,X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4)))
    <=> ( pp(aa(set(A),bool,finite_finite(A),A4))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4)) ) ) ).

% card_Diff1_less_iff
tff(fact_4907_card__Diff__singleton,axiom,
    ! [A: $tType,X: A,A4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),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,X),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_4908_card__Diff__singleton__if,axiom,
    ! [A: $tType,X: A,A4: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),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,X),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)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),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,X),bot_bot(set(A))))) = aa(set(A),nat,finite_card(A),A4) ) ) ) ).

% card_Diff_singleton_if
tff(fact_4909_prod__le__power,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(A)
     => ! [A4: set(B),F2: fun(B,A),N: A,K: nat] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),N)) ) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),K))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N),K))) ) ) ) ) ).

% prod_le_power
tff(fact_4910_sum__bounded__above__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_field(A)
     => ! [A4: set(B),F2: fun(B,A),K6: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),K6),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A4))))) )
         => ( pp(aa(set(B),bool,finite_finite(B),A4))
           => ( ( A4 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),K6)) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_4911_sum__bounded__above__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & semiring_1(A) )
     => ! [A4: set(B),F2: fun(B,A),K6: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,I3)),K6)) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A4)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),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))),K6))) ) ) ) ).

% sum_bounded_above_strict
tff(fact_4912_card__insert__le__m1,axiom,
    ! [A: $tType,N: nat,Y2: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),Y2))),N)) ) ) ).

% card_insert_le_m1
tff(fact_4913_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jv(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))),N)) ) ) ) ).

% polyfun_roots_card
tff(fact_4914_prod__gen__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A),C2: A] :
          ( pp(aa(set(B),bool,finite_finite(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_li(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),S3)),one_one(nat)))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_li(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,finite_card(B),S3)) ) ) ) ) ) ).

% prod_gen_delta
tff(fact_4915_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
       => ( code_nat_of_integer(K) = zero_zero(nat) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
       => ( code_nat_of_integer(K) = aa(product_prod(code_integer,code_integer),nat,product_case_prod(code_integer,code_integer,nat,aTP_Lamp_lj(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_4916_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A4)))
       => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_lk(set(A),fun(nat,fun(list(A),bool)),A4),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ai(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_4917_relpowp__fun__conv,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y2))
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = X )
          & ( aa(nat,A,F5,N) = Y2 )
          & ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,F5,I2)),aa(nat,A,F5,aa(nat,nat,suc,I2)))) ) ) ) ).

% relpowp_fun_conv
tff(fact_4918_relpowp__1,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),one_one(nat)),P) = P ).

% relpowp_1
tff(fact_4919_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_lk(set(A),fun(nat,fun(list(A),bool)),A4),N)))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4920_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_4921_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_4922_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
         => ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( ( I2 != J3 )
               => ( aa(nat,A,nth(A,Xs),I2) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_4923_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I: nat,J2: nat] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Xs),J2) )
          <=> ( I = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_4924_relpowp__Suc__E,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z))
     => ~ ! [Y3: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y3))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y3),Z)) ) ) ).

% relpowp_Suc_E
tff(fact_4925_relpowp__Suc__I,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y2: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y2))
     => ( pp(aa(A,bool,aa(A,fun(A,bool),P,Y2),Z))
       => pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z)) ) ) ).

% relpowp_Suc_I
tff(fact_4926_relpowp__Suc__D2,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z))
     => ? [Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y3))
          & pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),Y3),Z)) ) ) ).

% relpowp_Suc_D2
tff(fact_4927_relpowp__Suc__E2,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z))
     => ~ ! [Y3: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y3))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),Y3),Z)) ) ) ).

% relpowp_Suc_E2
tff(fact_4928_relpowp__Suc__I2,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y2: A,N: nat,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y2))
     => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),Y2),Z))
       => pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z)) ) ) ).

% relpowp_Suc_I2
tff(fact_4929_relpowp__0__I,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A] : pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P),X),X)) ).

% relpowp_0_I
tff(fact_4930_relpowp__0__E,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P),X),Y2))
     => ( X = Y2 ) ) ).

% relpowp_0_E
tff(fact_4931_relpowp_Osimps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),R2) = fequal(A) ).

% relpowp.simps(1)
tff(fact_4932_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ? [X2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs)))
            & ( aa(nat,A,nth(A,Xs),X2) = X )
            & ! [Y: nat] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(list(A),nat,size_size(list(A)),Xs)))
                  & ( aa(nat,A,nth(A,Xs),Y) = X ) )
               => ( Y = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4933_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A4: set(nat),B5: set(A)] :
      ( distinct(A,Xs)
     => ( ( A4 = set_ord_lessThan(nat,aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B5 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A4,B5) ) ) ) ).

% bij_betw_nth
tff(fact_4934_relpowp__E2,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y3: A,M3: nat] :
              ( ( N = aa(nat,nat,suc,M3) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y3))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M3),P),Y3),Z)) ) ) ) ) ).

% relpowp_E2
tff(fact_4935_relpowp__E,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y3: A,M3: nat] :
              ( ( N = aa(nat,nat,suc,M3) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M3),P),X),Y3))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y3),Z)) ) ) ) ) ).

% relpowp_E
tff(fact_4936_relpowp__bot,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),bot_bot(fun(A,fun(A,bool)))) = bot_bot(fun(A,fun(A,bool))) ) ) ).

% relpowp_bot
tff(fact_4937_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A4: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(set(A),nat,finite_card(A),A4)))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(set(A),fun(list(A),bool),aTP_Lamp_ll(nat,fun(set(A),fun(list(A),bool)),K),A4))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ai(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_4938_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,N,X)) = aa(set(A),set(A),insert(A,X),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),N)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_4939_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_4940_finite__enumerate,axiom,
    ! [S3: set(nat)] :
      ( pp(aa(set(nat),bool,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)))
          & ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(set(nat),nat,finite_card(nat),S3)))
             => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,R3,N4)),S3)) ) ) ) ).

% finite_enumerate
tff(fact_4941_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I: nat,X: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,I,X)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_4942_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I))
     => ( list_update(A,Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4943_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_4944_set__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),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),J2)),J2,aa(nat,A,nth(A,Xs),I))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_4945_distinct__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),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),J2)),J2,aa(nat,A,nth(A,Xs),I)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_4946_set__update__memI,axiom,
    ! [A: $tType,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),list_update(A,Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4947_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J2: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( ( I = J2 )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I,X)),J2) = X ) )
        & ( ( I != J2 )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I,X)),J2) = aa(nat,A,nth(A,Xs),J2) ) ) ) ) ).

% nth_list_update
tff(fact_4948_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( list_update(A,Xs,I,X) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_4949_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A3: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),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_4950_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,S5: A] :
              ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),R5),A4))
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S5),A4))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R5),S5)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,R5)),aa(A,B,F2,S5))) ) ) ) ).

% strict_mono_on_def
tff(fact_4951_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [R3: A,S2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),R3),A4))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S2),A4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R3),S2))
                 => pp(aa(B,bool,aa(B,fun(B,bool),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_4952_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)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),R),A4))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S),A4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R),S))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,R)),aa(A,B,F2,S))) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_4953_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,X,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_4954_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or5935395276787703475ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_4955_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_lb(nat,fun(nat,bool)),aTP_Lamp_cv(nat,fun(nat,bool))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_4956_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or5935395276787703475ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),U)) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_4957_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),set_or5935395276787703475ssThan(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ioo_iff
tff(fact_4958_semilattice__neutr__order_Oneutr__eq__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => ( ( Z = aa(A,A,aa(A,fun(A,A),F2,A3),B2) )
      <=> ( ( A3 = Z )
          & ( B2 = Z ) ) ) ) ).

% semilattice_neutr_order.neutr_eq_iff
tff(fact_4959_semilattice__neutr__order_Oeq__neutr__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = Z )
      <=> ( ( A3 = Z )
          & ( B2 = Z ) ) ) ) ).

% semilattice_neutr_order.eq_neutr_iff
tff(fact_4960_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite(A),set_or5935395276787703475ssThan(A,A3,B2))) ) ) ).

% infinite_Ioo
tff(fact_4961_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_4962_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_4963_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_4964_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_4965_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_4966_length__remove1,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% length_remove1
tff(fact_4967_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_lm(nat,fun(nat,bool))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_4968_set__removeAll,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),removeAll(A,X,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_4969_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_4970_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,int,code_int_of_integer,K) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K))) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( ( ( K = zero_zero(code_integer) )
           => ( aa(code_integer,int,code_int_of_integer,K) = zero_zero(int) ) )
          & ( ( K != zero_zero(code_integer) )
           => ( aa(code_integer,int,code_int_of_integer,K) = aa(product_prod(code_integer,code_integer),int,product_case_prod(code_integer,code_integer,int,aTP_Lamp_ln(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_4971_int__of__integer__of__nat,axiom,
    ! [N: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,semiring_1_of_nat(code_integer),N)) = aa(nat,int,semiring_1_of_nat(int),N) ).

% int_of_integer_of_nat
tff(fact_4972_zero__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,zero_zero(code_integer)) = zero_zero(int) ).

% zero_integer.rep_eq
tff(fact_4973_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),aa(nat,nat,suc,L)) ).

% card_greaterThanLessThan
tff(fact_4974_uminus__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),X)) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,X)) ).

% uminus_integer.rep_eq
tff(fact_4975_one__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_4976_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),X),Xa)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa)) ).

% minus_integer.rep_eq
tff(fact_4977_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),X),Xa)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa)) ).

% divide_integer.rep_eq
tff(fact_4978_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Xa))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa))) ) ).

% less_integer.rep_eq
tff(fact_4979_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),L))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L))) ) ).

% integer_less_iff
tff(fact_4980_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,L),U) = set_or5935395276787703475ssThan(nat,L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_4981_length__removeAll__less__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,X,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_removeAll_less_eq
tff(fact_4982_tanh__real__bounds,axiom,
    ! [X: real] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,tanh(real),X)),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))) ).

% tanh_real_bounds
tff(fact_4983_length__removeAll__less,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,X,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_removeAll_less
tff(fact_4984_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S3: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( ( S3 != bot_bot(set(A)) )
           => ~ ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
                  & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S3)))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4985_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = 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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% times_int.abs_eq
tff(fact_4986_Gcd__remove0__nat,axiom,
    ! [M10: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),M10))
     => ( gcd_Gcd(nat,M10) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M10),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_4987_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_4988_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( gcd_Gcd(A,A4) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A))))) ) ) ).

% Gcd_0_iff
tff(fact_4989_int_Oabs__induct,axiom,
    ! [P: fun(int,bool),X: int] :
      ( ! [Y3: product_prod(nat,nat)] : pp(aa(int,bool,P,aa(product_prod(nat,nat),int,abs_Integ,Y3)))
     => pp(aa(int,bool,P,X)) ) ).

% int.abs_induct
tff(fact_4990_Gcd__1,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),A4))
         => ( gcd_Gcd(A,A4) = one_one(A) ) ) ) ).

% Gcd_1
tff(fact_4991_Gcd__nat__eq__one,axiom,
    ! [N6: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),one_one(nat)),N6))
     => ( gcd_Gcd(nat,N6) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_4992_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X2: nat,Y3: nat] : Z != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X2),Y3)) ).

% eq_Abs_Integ
tff(fact_4993_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A4: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
           => ( gcd_Gcd(A,A4) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_4994_nat_Oabs__eq,axiom,
    ! [X: product_prod(nat,nat)] : aa(int,nat,nat2,aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),nat,product_case_prod(nat,nat,nat,minus_minus(nat)),X) ).

% nat.abs_eq
tff(fact_4995_zero__int__def,axiom,
    zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))) ).

% zero_int_def
tff(fact_4996_int__def,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),N) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,N),zero_zero(nat))) ).

% int_def
tff(fact_4997_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_lq(nat,fun(nat,product_prod(nat,nat)))),X)) ).

% uminus_int.abs_eq
tff(fact_4998_one__int__def,axiom,
    one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))) ).

% one_int_def
tff(fact_4999_of__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: product_prod(nat,nat)] : aa(int,A,ring_1_of_int(A),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_lr(nat,fun(nat,A))),X) ) ).

% of_int.abs_eq
tff(fact_5000_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lt(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X)) ) ).

% less_int.abs_eq
tff(fact_5001_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lv(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X)) ) ).

% less_eq_int.abs_eq
tff(fact_5002_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = 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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% plus_int.abs_eq
tff(fact_5003_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = 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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% minus_int.abs_eq
tff(fact_5004_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,collect(nat,aTP_Lamp_ma(nat,bool))) ) ).

% semiring_char_def
tff(fact_5005_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = inc(num_of_nat(N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = one2 ) ) ) ).

% num_of_nat.simps(2)
tff(fact_5006_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_5007_num__of__nat__numeral__eq,axiom,
    ! [Q5: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q5)) = Q5 ).

% num_of_nat_numeral_eq
tff(fact_5008_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_5009_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_5010_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_5011_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_5012_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,Y2: A] :
          ( ( X = Y2 )
        <=> ring_1_iszero(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) ) ) ).

% eq_iff_iszero_diff
tff(fact_5013_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_5014_Gcd__int__greater__eq__0,axiom,
    ! [K6: set(int)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K6))) ).

% Gcd_int_greater_eq_0
tff(fact_5015_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_5016_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y2: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y2) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y2)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_5017_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_5018_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_5019_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_5020_eq__numeral__iff__iszero_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y2: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(num,A,numeral_numeral(A),Y2) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,X,Y2)) ) ) ).

% eq_numeral_iff_iszero(1)
tff(fact_5021_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_5022_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_5023_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y2: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y2)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y2)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_5024_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_5025_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),one_one(nat)))
     => ( num_of_nat(N) = one2 ) ) ).

% num_of_nat_One
tff(fact_5026_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y2: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y2)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y2))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_5027_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y2: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),Y2) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y2))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_5028_eq__numeral__iff__iszero_I4_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y2: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y2)) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Y2,X)) ) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_5029_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ) ) ) ).

% numeral_num_of_nat_unfold
tff(fact_5030_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = aa(num,num,bit0,num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_5031_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M2)),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_5032_eq__numeral__iff__iszero_I6_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y2: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Y2) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,one2,Y2)) ) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_5033_eq__numeral__iff__iszero_I5_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = one_one(A) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,X,one2)) ) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_5034_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y2: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y2)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y2))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_5035_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lv(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa))) ) ).

% less_eq_int.rep_eq
tff(fact_5036_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),Xa))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lt(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa))) ) ).

% less_int.rep_eq
tff(fact_5037_prod__encode__def,axiom,
    nat_prod_encode = product_case_prod(nat,nat,nat,aTP_Lamp_mb(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_5038_prod__encode__eq,axiom,
    ! [X: product_prod(nat,nat),Y2: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),nat,nat_prod_encode,X) = aa(product_prod(nat,nat),nat,nat_prod_encode,Y2) )
    <=> ( X = Y2 ) ) ).

% prod_encode_eq
tff(fact_5039_le__prod__encode__1,axiom,
    ! [A3: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),B2)))) ).

% le_prod_encode_1
tff(fact_5040_le__prod__encode__2,axiom,
    ! [B2: nat,A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),B2)))) ).

% le_prod_encode_2
tff(fact_5041_nat_Orep__eq,axiom,
    ! [X: int] : aa(int,nat,nat2,X) = aa(product_prod(nat,nat),nat,product_case_prod(nat,nat,nat,minus_minus(nat)),aa(int,product_prod(nat,nat),rep_Integ,X)) ).

% nat.rep_eq
tff(fact_5042_of__int_Orep__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int] : aa(int,A,ring_1_of_int(A),X) = aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_lr(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,X)) ) ).

% of_int.rep_eq
tff(fact_5043_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M2: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M2) ).

% prod_encode_prod_decode_aux
tff(fact_5044_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),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_lq(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_5045_prod_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P2: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),aa(set(B),set(B),insert(B,I),I5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),I5) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),aa(set(B),set(B),insert(B,I),I5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P2,I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),I5)) ) ) ) ) ) ).

% prod.insert'
tff(fact_5046_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( linord4507533701916653071of_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,X),bot_bot(set(A))))) = remove1(A,X,linord4507533701916653071of_set(A,A4)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5047_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_5048_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_5049_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),I5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),collect(B,aa(set(B),fun(B,bool),aTP_Lamp_mc(fun(B,A),fun(set(B),fun(B,bool)),G),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I5) ) ).

% prod.non_neutral'
tff(fact_5050_prod_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),T5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,G,X2) = one_one(A) ) )
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                 => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_5051_prod_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),T5: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,H,I3) = one_one(A) ) )
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),S3))
                 => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),T5) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_5052_prod_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),T5: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,G,X2) = one_one(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S3) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_5053_prod_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),T5: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,G,X2) = one_one(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T5) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_5054_prod_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,A),fun(B,bool)),I5),G))))
         => ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_md(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_5055_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P2: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P2),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),I5) = one_one(A) ) ) ) ) ).

% prod.G_def
tff(fact_5056_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),aa(nat,nat,suc,I))))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J2))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_5057_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_5058_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_5059_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_5060_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),insert(A,X),A4)) = linorder_insort_key(A,A,aTP_Lamp_me(A,A),X,linord4507533701916653071of_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,X),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5061_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J2: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J2))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5062_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),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_5063_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or3652927894154168847AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),U)) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_5064_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_5065_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_5066_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),set_or3652927894154168847AtMost(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ioc_iff
tff(fact_5067_length__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),linorder_insort_key(B,A,F2,X,Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ) ).

% length_insort
tff(fact_5068_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or3652927894154168847AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_greaterThanAtMost
tff(fact_5069_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,L),U) = set_or3652927894154168847AtMost(nat,L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_5070_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite(A),set_or3652927894154168847AtMost(A,A3,B2))) ) ) ).

% infinite_Ioc
tff(fact_5071_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_5072_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_5073_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_5074_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] : set_or3652927894154168847AtMost(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_5075_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
           => ( linord4507533701916653071of_set(A,A4) = linorder_insort_key(A,A,aTP_Lamp_me(A,A),X,linord4507533701916653071of_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,X),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5076_sum__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A4: set(B),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,finite_finite(B),A4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_mf(fun(B,bool),fun(B,A),P)),A4) = aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_5077_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y2: nat,X: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y2))
       => ( image(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),C2),set_or7035219750837199246ssThan(nat,X,Y2)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y2),C2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y2))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y2))
           => ( image(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),C2),set_or7035219750837199246ssThan(nat,X,Y2)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y2))
           => ( image(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),C2),set_or7035219750837199246ssThan(nat,X,Y2)) = bot_bot(set(nat)) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_5078_rat__floor__lemma,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),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))),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,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_5079_inf__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F2),G),X) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ).

% inf_apply
tff(fact_5080_inf__right__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),Y2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) ) ).

% inf_right_idem
tff(fact_5081_inf_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).

% inf.right_idem
tff(fact_5082_inf__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) ) ).

% inf_left_idem
tff(fact_5083_inf_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).

% inf.left_idem
tff(fact_5084_inf__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),X) = X ) ).

% inf_idem
tff(fact_5085_inf_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),A3) = A3 ) ).

% inf.idem
tff(fact_5086_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) ) ) ) ).

% le_inf_iff
tff(fact_5087_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% inf.bounded_iff
tff(fact_5088_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).

% inf_bot_right
tff(fact_5089_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).

% inf_bot_left
tff(fact_5090_minus__rat__cancel,axiom,
    ! [A3: int,B2: int] : aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),A3)),aa(int,int,uminus_uminus(int),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ).

% minus_rat_cancel
tff(fact_5091_bij__betw__Suc,axiom,
    ! [M10: set(nat),N6: set(nat)] :
      ( bij_betw(nat,nat,suc,M10,N6)
    <=> ( image(nat,nat,suc,M10) = N6 ) ) ).

% bij_betw_Suc
tff(fact_5092_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_5093_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_5094_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),Y2)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_5095_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),aa(A,A,uminus_uminus(A),X))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_5096_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),X) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_5097_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),X)) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_5098_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A3: A,B2: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),D2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D2),A3)) ) ).

% image_diff_atLeastAtMost
tff(fact_5099_image__uminus__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y2: A] : image(A,A,uminus_uminus(A),set_or1337092689740270186AtMost(A,X,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,uminus_uminus(A),Y2),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atLeastAtMost
tff(fact_5100_Diff__disjoint,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_5101_image__uminus__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y2: A] : image(A,A,uminus_uminus(A),set_or5935395276787703475ssThan(A,X,Y2)) = set_or5935395276787703475ssThan(A,aa(A,A,uminus_uminus(A),Y2),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_greaterThanLessThan
tff(fact_5102_Compl__disjoint2,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),A4) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_5103_Compl__disjoint,axiom,
    ! [A: $tType,A4: set(A)] : 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)),A4)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_5104_Diff__Compl,axiom,
    ! [A: $tType,A4: set(A),B5: 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),uminus_uminus(set(A)),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5) ).

% Diff_Compl
tff(fact_5105_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J2: nat] : image(nat,nat,suc,set_or1337092689740270186AtMost(nat,I,J2)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J2)) ).

% image_Suc_atLeastAtMost
tff(fact_5106_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J2: nat] : image(nat,nat,suc,set_or7035219750837199246ssThan(nat,I,J2)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J2)) ).

% image_Suc_atLeastLessThan
tff(fact_5107_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N6: set(nat),A4: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N6,A4)
        <=> ( image(nat,A,semiring_1_of_nat(A),N6) = A4 ) ) ) ).

% bij_betw_of_nat
tff(fact_5108_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A3: A,B2: A] : image(A,A,aTP_Lamp_mh(A,fun(A,A),D2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_5109_minus__rat,axiom,
    ! [A3: int,B2: int] : aa(rat,rat,uminus_uminus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),A3)),B2) ).

% minus_rat
tff(fact_5110_divide__rat,axiom,
    ! [A3: int,B2: int,C2: int,D2: int] : aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)) = aa(int,rat,aa(int,fun(int,rat),fract,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),C2)) ).

% divide_rat
tff(fact_5111_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or3652927894154168847AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_greaterThanAtMost_int
tff(fact_5112_floor__Fract,axiom,
    ! [A3: int,B2: int] : archim6421214686448440834_floor(rat,aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) ).

% floor_Fract
tff(fact_5113_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A,B2: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),C2),set_or3652927894154168847AtMost(A,A3,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_5114_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A,B2: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),C2),set_or7035219750837199246ssThan(A,A3,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)) ) ).

% image_diff_atLeastLessThan
tff(fact_5115_image__uminus__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y2: A] : image(A,A,uminus_uminus(A),set_or7035219750837199246ssThan(A,X,Y2)) = set_or3652927894154168847AtMost(A,aa(A,A,uminus_uminus(A),Y2),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atLeastLessThan
tff(fact_5116_image__uminus__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y2: A] : image(A,A,uminus_uminus(A),set_or3652927894154168847AtMost(A,X,Y2)) = set_or7035219750837199246ssThan(A,aa(A,A,uminus_uminus(A),Y2),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_greaterThanAtMost
tff(fact_5117_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5118_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D2))
         => ( image(A,A,aTP_Lamp_mi(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_5119_less__rat,axiom,
    ! [B2: int,D2: int,A3: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),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_5120_add__rat,axiom,
    ! [B2: int,D2: int,A3: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(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),C2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2)) ) ) ) ).

% add_rat
tff(fact_5121_le__rat,axiom,
    ! [B2: int,D2: int,A3: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),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)))) ) ) ) ).

% le_rat
tff(fact_5122_diff__rat,axiom,
    ! [B2: int,D2: int,A3: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),minus_minus(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),C2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2)) ) ) ) ).

% diff_rat
tff(fact_5123_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),T2: set(A)] : image(A,A,aTP_Lamp_mj(A,fun(A,A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,aTP_Lamp_mj(A,fun(A,A),A3),S)),image(A,A,aTP_Lamp_mj(A,fun(A,A),A3),T2)) ) ).

% translation_subtract_Int
tff(fact_5124_Int__Diff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C5)) ).

% Int_Diff
tff(fact_5125_Diff__Int2,axiom,
    ! [A: $tType,A4: set(A),C5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C5)),B5) ).

% Diff_Int2
tff(fact_5126_Diff__Diff__Int,axiom,
    ! [A: $tType,A4: set(A),B5: 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),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5) ).

% Diff_Diff_Int
tff(fact_5127_Diff__Int__distrib,axiom,
    ! [A: $tType,C5: set(A),A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),A4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),B5)) ).

% Diff_Int_distrib
tff(fact_5128_Diff__Int__distrib2,axiom,
    ! [A: $tType,A4: set(A),B5: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C5)) ).

% Diff_Int_distrib2
tff(fact_5129_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).

% inf.strict_coboundedI2
tff(fact_5130_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).

% inf.strict_coboundedI1
tff(fact_5131_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5132_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% inf.strict_boundedE
tff(fact_5133_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_5134_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb3
tff(fact_5135_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% less_infI2
tff(fact_5136_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% less_infI1
tff(fact_5137_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).

% inf.coboundedI2
tff(fact_5138_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).

% inf.coboundedI1
tff(fact_5139_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5140_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5141_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2)) ) ).

% inf.cobounded2
tff(fact_5142_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),A3)) ) ).

% inf.cobounded1
tff(fact_5143_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5144_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z))) ) ) ) ).

% inf_greatest
tff(fact_5145_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))) ) ) ) ).

% inf.boundedI
tff(fact_5146_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% inf.boundedE
tff(fact_5147_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) = Y2 ) ) ) ).

% inf_absorb2
tff(fact_5148_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) = X ) ) ) ).

% inf_absorb1
tff(fact_5149_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_5150_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb1
tff(fact_5151_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) = X ) ) ) ).

% le_iff_inf
tff(fact_5152_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y2: A] :
          ( ! [X2: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X2),Y3)),X2))
         => ( ! [X2: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X2),Y3)),Y3))
           => ( ! [X2: A,Y3: A,Z3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F2,Y3),Z3))) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) = aa(A,A,aa(A,fun(A,A),F2,X),Y2) ) ) ) ) ) ).

% inf_unique
tff(fact_5153_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) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% inf.orderI
tff(fact_5154_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.orderE
tff(fact_5155_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% le_infI2
tff(fact_5156_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% le_infI1
tff(fact_5157_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_5158_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2))) ) ) ) ).

% le_infI
tff(fact_5159_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2)) ) ) ) ).

% le_infE
tff(fact_5160_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),Y2)) ) ).

% inf_le2
tff(fact_5161_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),X)) ) ).

% inf_le1
tff(fact_5162_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),X)) ) ).

% inf_sup_ord(1)
tff(fact_5163_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),Y2)) ) ).

% inf_sup_ord(2)
tff(fact_5164_inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F2: fun(A,B),G: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F2),G),X3) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) ) ).

% inf_fun_def
tff(fact_5165_inf__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z)) ) ).

% inf_left_commute
tff(fact_5166_inf_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).

% inf.left_commute
tff(fact_5167_inf__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),X) ) ).

% inf_commute
tff(fact_5168_inf_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: 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),B2),A3) ) ).

% inf.commute
tff(fact_5169_inf__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z)) ) ).

% inf_assoc
tff(fact_5170_inf_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).

% inf.assoc
tff(fact_5171_inf__sup__aci_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),X) ) ).

% inf_sup_aci(1)
tff(fact_5172_inf__sup__aci_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z)) ) ).

% inf_sup_aci(2)
tff(fact_5173_inf__sup__aci_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z)) ) ).

% inf_sup_aci(3)
tff(fact_5174_inf__sup__aci_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) ) ).

% inf_sup_aci(4)
tff(fact_5175_eq__rat_I3_J,axiom,
    ! [A3: int,C2: int] : aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),A3) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),C2) ).

% eq_rat(3)
tff(fact_5176_diff__eq,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y2)) ) ).

% diff_eq
tff(fact_5177_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),B2)) = bot_bot(A) ) ).

% inf_cancel_left2
tff(fact_5178_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),B2)) = bot_bot(A) ) ).

% inf_cancel_left1
tff(fact_5179_zero__notin__Suc__image,axiom,
    ! [A4: set(nat)] : ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),image(nat,nat,suc,A4))) ).

% zero_notin_Suc_image
tff(fact_5180_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),T2: 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),T2)) = 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),T2)) ) ).

% translation_diff
tff(fact_5181_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B),B5: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),image(B,A,F2,A4)),image(B,A,F2,B5))),image(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5)))) ).

% image_diff_subset
tff(fact_5182_eq__rat_I2_J,axiom,
    ! [A3: int] : aa(int,rat,aa(int,fun(int,rat),fract,A3),zero_zero(int)) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

% eq_rat(2)
tff(fact_5183_Rat__induct__pos,axiom,
    ! [P: fun(rat,bool),Q5: rat] :
      ( ! [A6: int,B4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
         => pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A6),B4))) )
     => pp(aa(rat,bool,P,Q5)) ) ).

% Rat_induct_pos
tff(fact_5184_mult__rat__cancel,axiom,
    ! [C2: int,A3: int,B2: int] :
      ( ( C2 != zero_zero(int) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),times_times(int),C2),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).

% mult_rat_cancel
tff(fact_5185_eq__rat_I1_J,axiom,
    ! [B2: int,D2: int,A3: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( ( aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) = aa(int,rat,aa(int,fun(int,rat),fract,C2),D2) )
        <=> ( aa(int,int,aa(int,fun(int,int),times_times(int),A3),D2) = aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2) ) ) ) ) ).

% eq_rat(1)
tff(fact_5186_Fract__of__nat__eq,axiom,
    ! [K: nat] : aa(int,rat,aa(int,fun(int,rat),fract,aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) = aa(nat,rat,semiring_1_of_nat(rat),K) ).

% Fract_of_nat_eq
tff(fact_5187_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T2: set(A)] : image(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),image(A,A,aa(A,fun(A,A),plus_plus(A),A3),T2)) ) ).

% translation_Compl
tff(fact_5188_Diff__triv,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) = A4 ) ) ).

% Diff_triv
tff(fact_5189_Int__Diff__disjoint,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_5190_rat__number__collapse_I6_J,axiom,
    ! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,K),zero_zero(int)) = zero_zero(rat) ).

% rat_number_collapse(6)
tff(fact_5191_rat__number__collapse_I1_J,axiom,
    ! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),K) = zero_zero(rat) ).

% rat_number_collapse(1)
tff(fact_5192_Fract__coprime,axiom,
    ! [A3: int,B2: int] : aa(int,rat,aa(int,fun(int,rat),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))) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ).

% Fract_coprime
tff(fact_5193_Diff__eq,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) = 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)),B5)) ).

% Diff_eq
tff(fact_5194_One__rat__def,axiom,
    one_one(rat) = aa(int,rat,aa(int,fun(int,rat),fract,one_one(int)),one_one(int)) ).

% One_rat_def
tff(fact_5195_Fract__of__int__eq,axiom,
    ! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,K),one_one(int)) = aa(int,rat,ring_1_of_int(rat),K) ).

% Fract_of_int_eq
tff(fact_5196_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A4: set(A),F2: fun(nat,A),N: nat] :
      ( ( A4 = image(nat,A,F2,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N))) )
     => pp(aa(set(A),bool,finite_finite(A),A4)) ) ).

% nat_seg_image_imp_finite
tff(fact_5197_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
    <=> ? [N5: nat,F5: fun(nat,A)] : A4 = image(nat,A,F5,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N5))) ) ).

% finite_conv_nat_seg_image
tff(fact_5198_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),T2: set(A)] : image(A,A,aTP_Lamp_mj(A,fun(A,A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),image(A,A,aTP_Lamp_mj(A,fun(A,A),A3),S)),image(A,A,aTP_Lamp_mj(A,fun(A,A),A3),T2)) ) ).

% translation_subtract_diff
tff(fact_5199_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T2: set(A)] : image(A,A,aTP_Lamp_mj(A,fun(A,A),A3),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),image(A,A,aTP_Lamp_mj(A,fun(A,A),A3),T2)) ) ).

% translation_subtract_Compl
tff(fact_5200_Fract__of__int__quotient,axiom,
    ! [K: int,L: int] : aa(int,rat,aa(int,fun(int,rat),fract,K),L) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,ring_1_of_int(rat),K)),aa(int,rat,ring_1_of_int(rat),L)) ).

% Fract_of_int_quotient
tff(fact_5201_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y2))) ) ) ).

% inf_shunt
tff(fact_5202_Zero__rat__def,axiom,
    zero_zero(rat) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

% Zero_rat_def
tff(fact_5203_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),B5))) ) ).

% disjoint_eq_subset_Compl
tff(fact_5204_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),W)),one_one(int)) = aa(num,rat,numeral_numeral(rat),W) ).

% rat_number_collapse(3)
tff(fact_5205_rat__number__expand_I3_J,axiom,
    ! [K: num] : aa(num,rat,numeral_numeral(rat),K) = aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% rat_number_expand(3)
tff(fact_5206_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_5207_sum_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),G: fun(B,A),B5: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(B),fun(B,A),aTP_Lamp_mk(fun(B,A),fun(set(B),fun(B,A)),G),B5)),A4) ) ) ) ).

% sum.inter_restrict
tff(fact_5208_prod_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),G: fun(B,A),B5: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(B),fun(B,A),aTP_Lamp_ml(fun(B,A),fun(set(B),fun(B,A)),G),B5)),A4) ) ) ) ).

% prod.inter_restrict
tff(fact_5209_sum_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( ! [X2: B,Y3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
               => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),A4))
                 => ( ( X2 != Y3 )
                   => ( ( aa(B,C,H,X2) = aa(B,C,H,Y3) )
                     => ( aa(C,A,G,aa(B,C,H,X2)) = zero_zero(A) ) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),image(B,C,H,A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),comp(C,A,B,G),H)),A4) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_5210_prod_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( ! [X2: B,Y3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A4))
               => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),A4))
                 => ( ( X2 != Y3 )
                   => ( ( aa(B,C,H,X2) = aa(B,C,H,Y3) )
                     => ( aa(C,A,G,aa(B,C,H,X2)) = one_one(A) ) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),image(B,C,H,A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),comp(C,A,B,G),H)),A4) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_5211_sum_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,finite_finite(B),S3))
           => ( ! [I3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,H,I3) = zero_zero(A) ) )
             => ( ! [I3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),T5)))
                   => ( aa(B,A,G,I3) = zero_zero(A) ) )
               => ( ! [X2: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S3),T5)))
                     => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T5) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5212_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,K: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
           => ( 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,X),bot_bot(set(A)))) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
           => ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ) ) ).

% Iio_Int_singleton
tff(fact_5213_sum_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),G: fun(B,A),B5: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5))) ) ) ) ).

% sum.Int_Diff
tff(fact_5214_image__Suc__lessThan,axiom,
    ! [N: nat] : image(nat,nat,suc,set_ord_lessThan(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).

% image_Suc_lessThan
tff(fact_5215_image__Suc__atMost,axiom,
    ! [N: nat] : image(nat,nat,suc,set_ord_atMost(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).

% image_Suc_atMost
tff(fact_5216_prod_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),G: fun(B,A),B5: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5))) ) ) ) ).

% prod.Int_Diff
tff(fact_5217_prod_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T5: set(B),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T5))
         => ( pp(aa(set(B),bool,finite_finite(B),S3))
           => ( ! [I3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,H,I3) = one_one(A) ) )
             => ( ! [I3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),T5)))
                   => ( aa(B,A,G,I3) = one_one(A) ) )
               => ( ! [X2: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S3),T5)))
                     => ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T5) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_5218_card__Diff__subset__Int,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5)))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = 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),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5))) ) ) ).

% card_Diff_subset_Int
tff(fact_5219_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image(nat,nat,suc,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_5220_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_5221_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image(nat,nat,suc,set_ord_lessThan(nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_5222_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_atMost(nat,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image(nat,nat,suc,set_ord_atMost(nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_5223_Fract__less__zero__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int))) ) ) ).

% Fract_less_zero_iff
tff(fact_5224_zero__less__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3)) ) ) ).

% zero_less_Fract_iff
tff(fact_5225_Fract__less__one__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B2)) ) ) ).

% Fract_less_one_iff
tff(fact_5226_one__less__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),A3)) ) ) ).

% one_less_Fract_iff
tff(fact_5227_rat__number__collapse_I5_J,axiom,
    aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ).

% rat_number_collapse(5)
tff(fact_5228_Fract__add__one,axiom,
    ! [N: int,M2: int] :
      ( ( N != zero_zero(int) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N)),N) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,M2),N)),one_one(rat)) ) ) ).

% Fract_add_one
tff(fact_5229_sum_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mm(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).

% sum.If_cases
tff(fact_5230_prod_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mn(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),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),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).

% prod.If_cases
tff(fact_5231_sum__image__le,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(C),G: fun(A,B),F2: fun(C,A)] :
          ( pp(aa(set(C),bool,finite_finite(C),I5))
         => ( ! [I3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I3),I5))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G,aa(C,A,F2,I3)))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),image(C,A,F2,I5))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,A),fun(C,B),comp(A,B,C,G),F2)),I5))) ) ) ) ).

% sum_image_le
tff(fact_5232_zero__le__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3)) ) ) ).

% zero_le_Fract_iff
tff(fact_5233_Fract__le__zero__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int))) ) ) ).

% Fract_le_zero_iff
tff(fact_5234_sum__div__partition,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A4: set(B),F2: fun(B,A),B2: A] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_mo(fun(B,A),fun(A,fun(B,A)),F2),B2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,aa(A,fun(B,bool),aTP_Lamp_mp(fun(B,A),fun(A,fun(B,bool)),F2),B2))))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,aa(A,fun(B,bool),aTP_Lamp_mq(fun(B,A),fun(A,fun(B,bool)),F2),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_5235_Fract__le__one__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),B2)) ) ) ).

% Fract_le_one_iff
tff(fact_5236_one__le__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3)) ) ) ).

% one_le_Fract_iff
tff(fact_5237_rat__number__expand_I5_J,axiom,
    ! [K: num] : aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% rat_number_expand(5)
tff(fact_5238_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa(int,rat,aa(int,fun(int,rat),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_5239_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( image(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,X,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
               => ( image(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,X,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
               => ( image(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,X,Y2)) = bot_bot(set(A)) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_5240_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A,C2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( image(A,A,aTP_Lamp_mr(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,X,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( image(A,A,aTP_Lamp_mr(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,X,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2)) ) ) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
           => ( image(A,A,aTP_Lamp_mr(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,X,Y2)) = bot_bot(set(A)) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_5241_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M2: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_ms(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_ms(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_ms(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_5242_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M2: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mt(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mt(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = 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),M2),A3)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mt(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = 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),M2),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_5243_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M2: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mu(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mu(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mu(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_5244_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M2: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mv(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mv(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = 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),M2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mv(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = 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),M2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_5245_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)] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( pp(aa(set(B),bool,finite_finite(B),R2))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,G,S3)),R2))
             => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_mw(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),S3) = aa(set(B),C,aa(fun(B,C),fun(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_my(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S3),G),F2)),R2) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_5246_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys2: 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),Ys2)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys2)) = 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)),Ys2))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_5247_ring__1__class_Oof__int__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_of_int(A) = aa(fun(product_prod(nat,nat),A),fun(int,A),map_fun(int,product_prod(nat,nat),A,A,rep_Integ,id(A)),product_case_prod(nat,nat,A,aTP_Lamp_lr(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_5248_take__bit__numeral__minus__numeral__int,axiom,
    ! [M2: num,N: num] : bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = 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_mz(num,fun(num,int),M2)),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M2),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_5249_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_5250_id__funpow,axiom,
    ! [A: $tType,N: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),id(A)) = id(A) ).

% id_funpow
tff(fact_5251_take__bit__num__simps_I1_J,axiom,
    ! [M2: num] : bit_take_bit_num(zero_zero(nat),M2) = none(num) ).

% take_bit_num_simps(1)
tff(fact_5252_group__add__class_Ominus__comp__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).

% group_add_class.minus_comp_minus
tff(fact_5253_boolean__algebra__class_Ominus__comp__minus,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).

% boolean_algebra_class.minus_comp_minus
tff(fact_5254_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_5255_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_5256_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa(nat,nat,suc,N),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(2)
tff(fact_5257_Gcd__int__eq,axiom,
    ! [N6: set(nat)] : gcd_Gcd(int,image(nat,int,semiring_1_of_nat(int),N6)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N6)) ).

% Gcd_int_eq
tff(fact_5258_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: num,N: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M2),aa(num,A,numeral_numeral(A),N)) = 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),M2),N)) ) ).

% take_bit_numeral_numeral
tff(fact_5259_option_Osimps_I5_J,axiom,
    ! [B: $tType,A: $tType,F1: B,F22: fun(A,B),X23: A] : aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),aa(A,option(A),some(A),X23)) = aa(A,B,F22,X23) ).

% option.simps(5)
tff(fact_5260_option_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F22: fun(A,B)] : aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),none(A)) = F1 ).

% option.simps(4)
tff(fact_5261_option_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,B),Option: option(A)] : aa(B,C,H,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)) = aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),case_option(C,A),aa(B,C,H,F1)),aa(fun(A,B),fun(A,C),aTP_Lamp_na(fun(B,C),fun(fun(A,B),fun(A,C)),H),F22)),Option) ).

% option.case_distrib
tff(fact_5262_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : bit_take_bit_num(N,one2) = case_nat(option(num),none(num),aTP_Lamp_nb(nat,option(num)),N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_5263_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_5264_None__notin__image__Some,axiom,
    ! [A: $tType,A4: set(A)] : ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),none(A)),image(A,option(A),some(A),A4))) ).

% None_notin_image_Some
tff(fact_5265_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys2: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys2)))
     => ( 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)),Ys2)) ) ) ).

% length_shuffles
tff(fact_5266_less__int__def,axiom,
    ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lt(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_int_def
tff(fact_5267_less__eq__int__def,axiom,
    ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lv(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_eq_int_def
tff(fact_5268_nat__def,axiom,
    nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))) ).

% nat_def
tff(fact_5269_in__image__insert__iff,axiom,
    ! [A: $tType,B5: set(set(A)),X: A,A4: set(A)] :
      ( ! [C6: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C6),B5))
         => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),C6)) )
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A4),image(set(A),set(A),insert(A,X),B5)))
      <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
          & pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(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,X),bot_bot(set(A))))),B5)) ) ) ) ).

% in_image_insert_iff
tff(fact_5270_image__int__atLeastAtMost,axiom,
    ! [A3: nat,B2: nat] : image(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,A3,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastAtMost
tff(fact_5271_image__int__atLeastLessThan,axiom,
    ! [A3: nat,B2: nat] : image(nat,int,semiring_1_of_nat(int),set_or7035219750837199246ssThan(nat,A3,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastLessThan
tff(fact_5272_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: num] :
          ( ( bit_take_bit_num(M2,N) = none(num) )
         => ( bit_se2584673776208193580ke_bit(A,M2,aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5273_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image(int,int,aTP_Lamp_nc(int,fun(int,int),L),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_5274_take__bit__num__def,axiom,
    ! [N: nat,M2: num] :
      ( ( ( bit_se2584673776208193580ke_bit(nat,N,aa(num,nat,numeral_numeral(nat),M2)) = zero_zero(nat) )
       => ( bit_take_bit_num(N,M2) = none(num) ) )
      & ( ( bit_se2584673776208193580ke_bit(nat,N,aa(num,nat,numeral_numeral(nat),M2)) != zero_zero(nat) )
       => ( bit_take_bit_num(N,M2) = aa(num,option(num),some(num),num_of_nat(bit_se2584673776208193580ke_bit(nat,N,aa(num,nat,numeral_numeral(nat),M2)))) ) ) ) ).

% take_bit_num_def
tff(fact_5275_Gcd__int__def,axiom,
    ! [K6: set(int)] : gcd_Gcd(int,K6) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)),K6))) ).

% Gcd_int_def
tff(fact_5276_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U))
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = image(nat,int,semiring_1_of_nat(int),set_ord_lessThan(nat,aa(int,nat,nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_5277_and__minus__numerals_I3_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = 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(M2,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_5278_and__minus__numerals_I7_J,axiom,
    ! [N: num,M2: 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,N)))),aa(num,int,numeral_numeral(int),M2)) = 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(M2,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_5279_and__minus__numerals_I8_J,axiom,
    ! [N: num,M2: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M2)) = 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(M2,aa(num,num,bit0,N))) ).

% and_minus_numerals(8)
tff(fact_5280_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M2: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M2)) = 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(N,M2))) ).

% take_bit_num_simps(4)
tff(fact_5281_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M2: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit0,M2)) = 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_nd(num,option(num))),bit_take_bit_num(N,M2)) ).

% take_bit_num_simps(3)
tff(fact_5282_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M2: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit0,M2)) = 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_nd(num,option(num))),bit_take_bit_num(pred_numeral(R),M2)) ).

% take_bit_num_simps(6)
tff(fact_5283_and__minus__numerals_I4_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = 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(M2,aa(num,num,bit0,N))) ).

% and_minus_numerals(4)
tff(fact_5284_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_5285_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fFalse),aTP_Lamp_ne(A,bool)),Option)) ) ).

% option.disc_eq_case(2)
tff(fact_5286_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fTrue),aTP_Lamp_nf(A,bool)),Option)) ) ).

% option.disc_eq_case(1)
tff(fact_5287_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_and_not_num(one2,aa(num,num,bit1,N)) = none(num) ).

% and_not_num.simps(3)
tff(fact_5288_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M2: num] : bit_take_bit_num(N,aa(num,num,bit0,M2)) = case_nat(option(num),none(num),aTP_Lamp_ng(num,fun(nat,option(num)),M2),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_5289_case__optionE,axiom,
    ! [A: $tType,P: bool,Q: fun(A,bool),X: option(A)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),P),Q),X))
     => ( ( ( X = none(A) )
         => ~ pp(P) )
       => ~ ! [Y3: A] :
              ( ( X = aa(A,option(A),some(A),Y3) )
             => ~ pp(aa(A,bool,Q,Y3)) ) ) ) ).

% case_optionE
tff(fact_5290_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M2: num] : bit_take_bit_num(N,aa(num,num,bit1,M2)) = case_nat(option(num),none(num),aTP_Lamp_nh(num,fun(nat,option(num)),M2),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_5291_and__not__num__eq__None__iff,axiom,
    ! [M2: num,N: num] :
      ( ( bit_and_not_num(M2,N) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_5292_int__numeral__and__not__num,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = 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(M2,N)) ).

% int_numeral_and_not_num
tff(fact_5293_int__numeral__not__and__num,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M2))),aa(num,int,numeral_numeral(int),N)) = 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(N,M2)) ).

% int_numeral_not_and_num
tff(fact_5294_measure__function__int,axiom,
    fun_is_measure(int,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))) ).

% measure_function_int
tff(fact_5295_Some__image__these__eq,axiom,
    ! [A: $tType,A4: set(option(A))] : image(A,option(A),some(A),these(A,A4)) = collect(option(A),aTP_Lamp_ni(set(option(A)),fun(option(A),bool),A4)) ).

% Some_image_these_eq
tff(fact_5296_positive__rat,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(rat,bool,positive,aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B2))) ) ).

% positive_rat
tff(fact_5297_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_5298_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_5299_these__insert__Some,axiom,
    ! [A: $tType,X: A,A4: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),aa(A,option(A),some(A),X)),A4)) = aa(set(A),set(A),insert(A,X),these(A,A4)) ).

% these_insert_Some
tff(fact_5300_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_5301_in__these__eq,axiom,
    ! [A: $tType,X: A,A4: set(option(A))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),these(A,A4)))
    <=> pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),aa(A,option(A),some(A),X)),A4)) ) ).

% in_these_eq
tff(fact_5302_measure__size,axiom,
    ! [A: $tType] :
      ( size(A)
     => fun_is_measure(A,size_size(A)) ) ).

% measure_size
tff(fact_5303_these__not__empty__eq,axiom,
    ! [A: $tType,B5: set(option(A))] :
      ( ( these(A,B5) != bot_bot(set(A)) )
    <=> ( ( B5 != bot_bot(set(option(A))) )
        & ( B5 != aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_5304_these__empty__eq,axiom,
    ! [A: $tType,B5: set(option(A))] :
      ( ( these(A,B5) = bot_bot(set(A)) )
    <=> ( ( B5 = bot_bot(set(option(A))) )
        | ( B5 = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_5305_Rat_Opositive__minus,axiom,
    ! [X: rat] :
      ( ~ pp(aa(rat,bool,positive,X))
     => ( ( X != zero_zero(rat) )
       => pp(aa(rat,bool,positive,aa(rat,rat,uminus_uminus(rat),X))) ) ) ).

% Rat.positive_minus
tff(fact_5306_less__rat__def,axiom,
    ! [X: rat,Y2: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y2))
    <=> pp(aa(rat,bool,positive,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Y2),X))) ) ).

% less_rat_def
tff(fact_5307_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa(rat,bool,positive,X))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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,X))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_5308_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y2: option(num)] :
      ( ( bit_and_not_num(X,Xa) = Y2 )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y2 != none(num) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa = aa(num,num,bit0,N2)
             => ( Y2 != aa(num,option(num),some(num),one2) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa = aa(num,num,bit1,N2)
               => ( Y2 != none(num) ) ) )
           => ( ! [M3: num] :
                  ( ( X = aa(num,num,bit0,M3) )
                 => ( ( Xa = one2 )
                   => ( Y2 != aa(num,option(num),some(num),aa(num,num,bit0,M3)) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ! [N2: num] :
                        ( ( Xa = aa(num,num,bit0,N2) )
                       => ( Y2 != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M3,N2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit1,N2) )
                         => ( Y2 != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M3,N2)) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit1,M3) )
                       => ( ( Xa = one2 )
                         => ( Y2 != aa(num,option(num),some(num),aa(num,num,bit0,M3)) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ! [N2: num] :
                              ( ( Xa = aa(num,num,bit0,N2) )
                             => ( Y2 != 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_nj(num,option(num))),bit_and_not_num(M3,N2)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit1,N2) )
                               => ( Y2 != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M3,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_5309_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M2: num] : bit_take_bit_num(N,M2) = aa(product_prod(nat,num),option(num),product_case_prod(nat,num,option(num),aTP_Lamp_nn(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),product_Pair(nat,num,N),M2)) ).

% Bit_Operations.take_bit_num_code
tff(fact_5310_map__option__eq__Some,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xo: option(B),Y2: 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),Y2) )
    <=> ? [Z5: B] :
          ( ( Xo = aa(B,option(B),some(B),Z5) )
          & ( aa(B,A,F2,Z5) = Y2 ) ) ) ).

% map_option_eq_Some
tff(fact_5311_option_Omap__disc__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: option(A)] :
      ( ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),A3) = none(B) )
    <=> ( A3 = none(A) ) ) ).

% option.map_disc_iff
tff(fact_5312_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_5313_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: option(B)] :
      ( ( none(A) = aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),X) )
    <=> ( X = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_5314_case__map__option,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: A,H: fun(B,A),F2: fun(C,B),X: 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),X)) = 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)),X) ).

% case_map_option
tff(fact_5315_num_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(num,A),F32: fun(num,A),Num: num] : aa(A,B,H,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),Num)) = 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),aa(A,B,H,F1)),aa(fun(num,A),fun(num,B),aTP_Lamp_no(fun(A,B),fun(fun(num,A),fun(num,B)),H),F22)),aa(fun(num,A),fun(num,B),aTP_Lamp_no(fun(A,B),fun(fun(num,A),fun(num,B)),H),F32)),Num) ).

% num.case_distrib
tff(fact_5316_option_Omap__ident,axiom,
    ! [A: $tType,T2: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_np(A,A)),T2) = T2 ).

% option.map_ident
tff(fact_5317_option_Osimps_I9_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X23: A] : aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),aa(A,option(A),some(A),X23)) = aa(B,option(B),some(B),aa(A,B,F2,X23)) ).

% option.simps(9)
tff(fact_5318_map__option__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y2: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( X = Y2 )
     => ( ! [A6: A] :
            ( ( Y2 = aa(A,option(A),some(A),A6) )
           => ( aa(A,B,F2,A6) = aa(A,B,G,A6) ) )
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Y2) ) ) ) ).

% map_option_cong
tff(fact_5319_option_Osimps_I8_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B)] : aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),none(A)) = none(B) ).

% option.simps(8)
tff(fact_5320_map__option_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(B,C),G: fun(A,B),Option: option(A)] : aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),F2),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Option)) = aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,F2),G)),Option) ).

% map_option.compositionality
tff(fact_5321_option_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F2: fun(A,B),V2: option(A)] : aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),G),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),V2)) = aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),F2)),V2) ).

% option.map_comp
tff(fact_5322_map__option_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,C),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),option(C)),comp(option(B),option(C),option(A),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),F2)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G)) = aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,F2),G)) ).

% map_option.comp
tff(fact_5323_map__option_Oidentity,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_np(A,A)) = id(option(A)) ).

% map_option.identity
tff(fact_5324_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_5325_option_Omap__id,axiom,
    ! [A: $tType,T2: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),id(A)),T2) = T2 ).

% option.map_id
tff(fact_5326_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X23: 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,X23)) = aa(num,A,F22,X23) ).

% verit_eq_simplify(17)
tff(fact_5327_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_5328_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_5329_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_5330_map__option__case,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Y2: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),Y2) = 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_nq(fun(B,A),fun(B,option(A)),F2)),Y2) ).

% map_option_case
tff(fact_5331_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,B),X3: 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_nr(A,option(C))),X3) = none(B) ).

% map_option_o_empty
tff(fact_5332_Rat_Opositive__def,axiom,
    positive = aa(fun(product_prod(int,int),bool),fun(rat,bool),map_fun(rat,product_prod(int,int),bool,bool,rep_Rat,id(bool)),aTP_Lamp_ns(product_prod(int,int),bool)) ).

% Rat.positive_def
tff(fact_5333_and__num_Oelims,axiom,
    ! [X: num,Xa: num,Y2: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa) = Y2 )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y2 != aa(num,option(num),some(num),one2) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa = aa(num,num,bit0,N2)
             => ( Y2 != none(num) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa = aa(num,num,bit1,N2)
               => ( Y2 != aa(num,option(num),some(num),one2) ) ) )
           => ( ( ? [M3: num] : X = aa(num,num,bit0,M3)
               => ( ( Xa = one2 )
                 => ( Y2 != none(num) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ! [N2: num] :
                        ( ( Xa = aa(num,num,bit0,N2) )
                       => ( Y2 != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M3,N2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit1,N2) )
                         => ( Y2 != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M3,N2)) ) ) )
                 => ( ( ? [M3: num] : X = aa(num,num,bit1,M3)
                     => ( ( Xa = one2 )
                       => ( Y2 != aa(num,option(num),some(num),one2) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ! [N2: num] :
                              ( ( Xa = aa(num,num,bit0,N2) )
                             => ( Y2 != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M3,N2)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit1,N2) )
                               => ( Y2 != 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_nj(num,option(num))),bit_un7362597486090784418nd_num(M3,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_5334_and__num_Osimps_I4_J,axiom,
    ! [M2: num] : bit_un7362597486090784418nd_num(aa(num,num,bit0,M2),one2) = none(num) ).

% and_num.simps(4)
tff(fact_5335_and__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_un7362597486090784418nd_num(one2,aa(num,num,bit0,N)) = none(num) ).

% and_num.simps(2)
tff(fact_5336_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: num] :
          ( ( bit_un7362597486090784418nd_num(M2,N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_5337_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = 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_un7362597486090784418nd_num(M2,N)) ) ).

% numeral_and_num
tff(fact_5338_xor__num_Oelims,axiom,
    ! [X: num,Xa: num,Y2: option(num)] :
      ( ( bit_un2480387367778600638or_num(X,Xa) = Y2 )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y2 != none(num) ) ) )
       => ( ( ( X = one2 )
           => ! [N2: num] :
                ( ( Xa = aa(num,num,bit0,N2) )
               => ( Y2 != aa(num,option(num),some(num),aa(num,num,bit1,N2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa = aa(num,num,bit1,N2) )
                 => ( Y2 != aa(num,option(num),some(num),aa(num,num,bit0,N2)) ) ) )
           => ( ! [M3: num] :
                  ( ( X = aa(num,num,bit0,M3) )
                 => ( ( Xa = one2 )
                   => ( Y2 != aa(num,option(num),some(num),aa(num,num,bit1,M3)) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ! [N2: num] :
                        ( ( Xa = aa(num,num,bit0,N2) )
                       => ( Y2 != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un2480387367778600638or_num(M3,N2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit1,N2) )
                         => ( Y2 != 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_un2480387367778600638or_num(M3,N2))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit1,M3) )
                       => ( ( Xa = one2 )
                         => ( Y2 != aa(num,option(num),some(num),aa(num,num,bit0,M3)) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ! [N2: num] :
                              ( ( Xa = aa(num,num,bit0,N2) )
                             => ( Y2 != 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_un2480387367778600638or_num(M3,N2))) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit1,N2) )
                               => ( Y2 != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un2480387367778600638or_num(M3,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_5339_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_nt(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_5340_option_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,C),F2: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),C),comp(option(B),C,option(A),aa(fun(B,C),fun(option(B),C),aa(C,fun(fun(B,C),fun(option(B),C)),rec_option(C,B),G),Ga)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2)) = aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),G),aa(fun(A,B),fun(A,C),aTP_Lamp_na(fun(B,C),fun(fun(A,B),fun(A,C)),Ga),F2)) ).

% option.rec_o_map
tff(fact_5341_of__rat__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),zero_zero(rat)) = zero_zero(A) ) ) ).

% of_rat_0
tff(fact_5342_of__rat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( aa(rat,A,field_char_0_of_rat(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(rat) ) ) ) ).

% of_rat_eq_0_iff
tff(fact_5343_zero__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( zero_zero(A) = aa(rat,A,field_char_0_of_rat(A),A3) )
        <=> ( zero_zero(rat) = A3 ) ) ) ).

% zero_eq_of_rat_iff
tff(fact_5344_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_5345_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_5346_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_5347_of__rat__of__nat__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat] : aa(rat,A,field_char_0_of_rat(A),aa(nat,rat,semiring_1_of_nat(rat),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% of_rat_of_nat_eq
tff(fact_5348_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R),zero_zero(rat))) ) ) ).

% of_rat_less_0_iff
tff(fact_5349_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R)) ) ) ).

% zero_less_of_rat_iff
tff(fact_5350_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),R)) ) ) ).

% one_less_of_rat_iff
tff(fact_5351_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R),one_one(rat))) ) ) ).

% of_rat_less_1_iff
tff(fact_5352_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),zero_zero(rat))) ) ) ).

% of_rat_le_0_iff
tff(fact_5353_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),R)) ) ) ).

% zero_le_of_rat_iff
tff(fact_5354_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_5355_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R)) ) ) ).

% one_le_of_rat_iff
tff(fact_5356_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),one_one(rat))) ) ) ).

% of_rat_le_1_iff
tff(fact_5357_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_5358_of__rat__dense,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
     => ? [Q3: rat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(rat,real,field_char_0_of_rat(real),Q3)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(rat,real,field_char_0_of_rat(real),Q3)),Y2)) ) ) ).

% of_rat_dense
tff(fact_5359_of__rat__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat,S: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R)),aa(rat,A,field_char_0_of_rat(A),S)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R),S)) ) ) ).

% of_rat_less
tff(fact_5360_of__rat__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),A3)) = aa(A,A,uminus_uminus(A),aa(rat,A,field_char_0_of_rat(A),A3)) ) ).

% of_rat_minus
tff(fact_5361_of__rat__diff,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),minus_minus(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(rat,A,field_char_0_of_rat(A),A3)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_diff
tff(fact_5362_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_5363_option_Osimps_I7_J,axiom,
    ! [C: $tType,A: $tType,F1: C,F22: fun(A,C),X23: A] : aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),F1),F22),aa(A,option(A),some(A),X23)) = aa(A,C,F22,X23) ).

% option.simps(7)
tff(fact_5364_option_Osimps_I6_J,axiom,
    ! [A: $tType,C: $tType,F1: C,F22: fun(A,C)] : aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),F1),F22),none(A)) = F1 ).

% option.simps(6)
tff(fact_5365_xor__num_Osimps_I1_J,axiom,
    bit_un2480387367778600638or_num(one2,one2) = none(num) ).

% xor_num.simps(1)
tff(fact_5366_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_5367_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),aa(int,rat,aa(int,fun(int,rat),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_5368_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: num] :
          ( ( bit_un2480387367778600638or_num(M2,N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_5369_of__rat_Orep__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [X: rat] : aa(rat,A,field_char_0_of_rat(A),X) = 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,X)))),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,X)))) ) ).

% of_rat.rep_eq
tff(fact_5370_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = 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_un2480387367778600638or_num(M2,N)) ) ).

% numeral_xor_num
tff(fact_5371_inverse__rat__def,axiom,
    inverse_inverse(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_nu(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_5372_uminus__rat__def,axiom,
    uminus_uminus(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_nv(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat_def
tff(fact_5373_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_5374_take__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( take(A,N,Xs) = Xs ) ) ).

% take_all
tff(fact_5375_take__all__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( take(A,N,Xs) = Xs )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% take_all_iff
tff(fact_5376_nth__take,axiom,
    ! [A: $tType,I: nat,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
     => ( aa(nat,A,nth(A,take(A,N,Xs)),I) = aa(nat,A,nth(A,Xs),I) ) ) ).

% nth_take
tff(fact_5377_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys2: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys2)))
       => ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K))
             => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys2),I3) ) )
         => ( take(A,K,Xs) = take(A,K,Ys2) ) ) ) ) ).

% nth_take_lemma
tff(fact_5378_one__rat__def,axiom,
    one_one(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))) ).

% one_rat_def
tff(fact_5379_Fract_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa(int,rat,aa(int,fun(int,rat),fract,Xa),X) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),X),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,Xa),X))) ).

% Fract.abs_eq
tff(fact_5380_zero__rat__def,axiom,
    zero_zero(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int))) ).

% zero_rat_def
tff(fact_5381_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),X)),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),X)),aa(product_prod(int,int),int,product_fst(int,int),X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_5382_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( pp(aa(rat,bool,positive,aa(product_prod(int,int),rat,abs_Rat,X)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),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),X)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ).

% Rat.positive.abs_eq
tff(fact_5383_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( aa(rat,rat,uminus_uminus(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(product_prod(int,int),int,product_snd(int,int),X))) ) ) ).

% uminus_rat.abs_eq
tff(fact_5384_ratrel__iff,axiom,
    ! [X: product_prod(int,int),Y2: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),Y2))
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Y2) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Y2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Y2)),aa(product_prod(int,int),int,product_snd(int,int),X)) ) ) ) ).

% ratrel_iff
tff(fact_5385_one__rat_Orsp,axiom,
    pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))),aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int)))) ).

% one_rat.rsp
tff(fact_5386_zero__rat_Orsp,axiom,
    pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int))),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)))) ).

% zero_rat.rsp
tff(fact_5387_ratrel__def,axiom,
    ! [X3: product_prod(int,int),Xa2: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X3),Xa2))
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X3) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Xa2) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X3)),aa(product_prod(int,int),int,product_snd(int,int),Xa2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X3)) ) ) ) ).

% ratrel_def
tff(fact_5388_of__rat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [X: product_prod(int,int)] :
          ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
         => ( aa(rat,A,field_char_0_of_rat(A),aa(product_prod(int,int),rat,abs_Rat,X)) = 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),X))),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),X))) ) ) ) ).

% of_rat.abs_eq
tff(fact_5389_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2)),lex(A,R)))
     => ~ ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Ys2)))
             => ( ( take(A,I3,Xs) = take(A,I3,Ys2) )
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Ys2),I3))),R)) ) ) ) ) ).

% lex_take_index
tff(fact_5390_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
       => ( code_num_of_integer(K) = one2 ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
       => ( code_num_of_integer(K) = aa(product_prod(code_integer,code_integer),num,product_case_prod(code_integer,code_integer,num,aTP_Lamp_nw(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_5391_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat,M2: 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))),N)),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))),M2)) = 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),M2),N))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_5392_min_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),A3) = A3 ) ).

% min.idem
tff(fact_5393_min_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).

% min.left_idem
tff(fact_5394_min_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).

% min.right_idem
tff(fact_5395_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% min.bounded_iff
tff(fact_5396_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_5397_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb1
tff(fact_5398_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb3
tff(fact_5399_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_5400_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),Y2)) ) ) ) ).

% min_less_iff_conj
tff(fact_5401_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N)) ).

% min_Suc_Suc
tff(fact_5402_min__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% min_0L
tff(fact_5403_min__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_5404_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) ) ) ) ).

% min_number_of(1)
tff(fact_5405_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_5406_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_5407_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_5408_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_5409_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_5410_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_5411_length__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),take(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_take
tff(fact_5412_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( 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),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( 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),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) ) ) ) ).

% min_number_of(4)
tff(fact_5413_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( 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),V2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( 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),V2)) = aa(num,A,numeral_numeral(A),V2) ) ) ) ) ).

% min_number_of(3)
tff(fact_5414_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( 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),V2))) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( 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),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) ) ) ) ).

% min_number_of(2)
tff(fact_5415_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K)),N)) ).

% min_numeral_Suc
tff(fact_5416_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_5417_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% min_add_distrib_right
tff(fact_5418_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y2: 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),X),Y2)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y2),Z)) ) ).

% min_add_distrib_left
tff(fact_5419_min_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).

% min.assoc
tff(fact_5420_min_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: 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),B2),A3) ) ).

% min.commute
tff(fact_5421_min_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),B2),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).

% min.left_commute
tff(fact_5422_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Xa2) = X3 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Xa2) = Xa2 ) ) ) ) ).

% min_def_raw
tff(fact_5423_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_5424_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.orderE
tff(fact_5425_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) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% min.orderI
tff(fact_5426_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% min.boundedE
tff(fact_5427_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))) ) ) ) ).

% min.boundedI
tff(fact_5428_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5429_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3)) ) ).

% min.cobounded1
tff(fact_5430_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2)) ) ).

% min.cobounded2
tff(fact_5431_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5432_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5433_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).

% min.coboundedI1
tff(fact_5434_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).

% min.coboundedI2
tff(fact_5435_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Z)) ) ) ) ).

% min_le_iff_disj
tff(fact_5436_of__int__min,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Y2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_min(int),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(int,A,ring_1_of_int(A),X)),aa(int,A,ring_1_of_int(A),Y2)) ) ).

% of_int_min
tff(fact_5437_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),Z)) ) ) ) ).

% min_less_iff_disj
tff(fact_5438_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% min.strict_boundedE
tff(fact_5439_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5440_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).

% min.strict_coboundedI1
tff(fact_5441_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).

% min.strict_coboundedI2
tff(fact_5442_max__min__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),A3)) ) ).

% max_min_distrib1
tff(fact_5443_max__min__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) ) ).

% max_min_distrib2
tff(fact_5444_min__max__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),A3)) ) ).

% min_max_distrib1
tff(fact_5445_min__max__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(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),A3),C2)) ) ).

% min_max_distrib2
tff(fact_5446_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y2: 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),X),Y2)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y2),Z)) ) ).

% min_diff_distrib_left
tff(fact_5447_min__diff,axiom,
    ! [M2: nat,I: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),I)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N)),I) ).

% min_diff
tff(fact_5448_nat__mult__min__left,axiom,
    ! [M2: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N)),Q5) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)) ).

% nat_mult_min_left
tff(fact_5449_nat__mult__min__right,axiom,
    ! [M2: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q5)) ).

% nat_mult_min_right
tff(fact_5450_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y2: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y2)) ) ).

% of_nat_min
tff(fact_5451_inf__min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( inf_inf(A) = ord_min(A) ) ) ).

% inf_min
tff(fact_5452_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_5453_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y2)) ) ).

% minus_min_eq_max
tff(fact_5454_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y2)) ) ).

% minus_max_eq_min
tff(fact_5455_concat__bit__assoc__sym,axiom,
    ! [M2: nat,N: nat,K: int,L: int,R: int] : bit_concat_bit(M2,bit_concat_bit(N,K,L),R) = bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N),K,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),L,R)) ).

% concat_bit_assoc_sym
tff(fact_5456_take__bit__concat__bit__eq,axiom,
    ! [M2: nat,N: nat,K: int,L: int] : bit_se2584673776208193580ke_bit(int,M2,bit_concat_bit(N,K,L)) = bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N),K,bit_se2584673776208193580ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),L)) ).

% take_bit_concat_bit_eq
tff(fact_5457_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),P2)) ) ) ) ) ).

% min_mult_distrib_right
tff(fact_5458_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),P2)) ) ) ) ) ).

% max_mult_distrib_right
tff(fact_5459_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y2)) ) ) ) ) ).

% min_mult_distrib_left
tff(fact_5460_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y2)) ) ) ) ) ).

% max_mult_distrib_left
tff(fact_5461_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y2),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y2),P2)) ) ) ) ) ).

% max_divide_distrib_right
tff(fact_5462_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y2),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y2),P2)) ) ) ) ) ).

% min_divide_distrib_right
tff(fact_5463_min__Suc1,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),M2) = case_nat(nat,zero_zero(nat),aTP_Lamp_nx(nat,fun(nat,nat),N),M2) ).

% min_Suc1
tff(fact_5464_min__Suc2,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_ny(nat,fun(nat,nat),N),M2) ).

% min_Suc2
tff(fact_5465_lexord__take__index__conv,axiom,
    ! [A: $tType,X: list(A),Y2: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Y2)),lexord(A,R)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y2)))
          & ( take(A,aa(list(A),nat,size_size(list(A)),X),Y2) = X ) )
        | ? [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y2))))
            & ( take(A,I2,X) = take(A,I2,Y2) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,X),I2)),aa(nat,A,nth(A,Y2),I2))),R)) ) ) ) ).

% lexord_take_index_conv
tff(fact_5466_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2)),listrel1(A,R)))
    <=> ? [Y4: A,N5: nat] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),N5)),Y4)),R))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( Ys2 = list_update(A,Xs,N5,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_5467_lenlex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_nz(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% lenlex_conv
tff(fact_5468_inf__int__def,axiom,
    inf_inf(int) = ord_min(int) ).

% inf_int_def
tff(fact_5469_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2)),listrel1(A,R)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) ) ) ).

% listrel1_eq_len
tff(fact_5470_lexord__lex,axiom,
    ! [A: $tType,X: list(A),Y2: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Y2)),lex(A,R)))
    <=> ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Y2)),lexord(A,R)))
        & ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y2) ) ) ) ).

% lexord_lex
tff(fact_5471_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ms),Ns)),lenlex(A,R)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),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_5472_rcis__cnj,axiom,
    ! [A3: complex] : cnj(A3) = rcis(real_V7770717601297561774m_norm(complex,A3),aa(real,real,uminus_uminus(real),arg(A3))) ).

% rcis_cnj
tff(fact_5473_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),X) )
    <=> ? [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I2)))
          & ( X = aa(nat,A,nth(A,Xs),I2) )
          & ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I2))
             => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_5474_find__Some__iff2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),X) = find(A,P,Xs) )
    <=> ? [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I2)))
          & ( X = aa(nat,A,nth(A,Xs),I2) )
          & ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I2))
             => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_5475_cis__rcis__eq,axiom,
    ! [A3: real] : cis(A3) = rcis(one_one(real),A3) ).

% cis_rcis_eq
tff(fact_5476_find__None__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( find(A,P,Xs) = none(A) )
    <=> ~ ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(A,bool,P,X4)) ) ) ).

% find_None_iff
tff(fact_5477_find__None__iff2,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( none(A) = find(A,P,Xs) )
    <=> ~ ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(A,bool,P,X4)) ) ) ).

% find_None_iff2
tff(fact_5478_rcis__divide,axiom,
    ! [R1: real,A3: real,R22: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),rcis(R1,A3)),rcis(R22,B2)) = rcis(aa(real,real,aa(real,fun(real,real),divide_divide(real),R1),R22),aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),B2)) ).

% rcis_divide
tff(fact_5479_rcis__inverse,axiom,
    ! [R: real,A3: real] : aa(complex,complex,inverse_inverse(complex),rcis(R,A3)) = rcis(aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),R),aa(real,real,uminus_uminus(real),A3)) ).

% rcis_inverse
tff(fact_5480_nth__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A),M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,M2),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_5481_nth__zip,axiom,
    ! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys2)))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys2)),I) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_5482_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the(product_prod(int,int),aTP_Lamp_oa(rat,fun(product_prod(int,int),bool),X)) ).

% quotient_of_def
tff(fact_5483_coprime__minus__left__iff,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,uminus_uminus(A),A3),B2)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_minus_left_iff
tff(fact_5484_coprime__minus__right__iff,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,aa(A,A,uminus_uminus(A),B2))
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_minus_right_iff
tff(fact_5485_rotate0,axiom,
    ! [A: $tType] : rotate(A,zero_zero(nat)) = id(list(A)) ).

% rotate0
tff(fact_5486_length__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate(A,N),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate
tff(fact_5487_coprime__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,A3)
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% coprime_self
tff(fact_5488_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_5489_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_5490_coprime__power__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,N: nat] :
          ( algebr8660921524188924756oprime(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
        <=> ( algebr8660921524188924756oprime(A,A3,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_5491_coprime__power__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,N: nat,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),B2)
        <=> ( algebr8660921524188924756oprime(A,A3,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_5492_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_5493_rotate__Suc,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,aa(nat,nat,suc,N)),Xs) = rotate1(A,aa(list(A),list(A),rotate(A,N),Xs)) ).

% rotate_Suc
tff(fact_5494_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,zero_zero(A))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% coprime_0_right_iff
tff(fact_5495_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A3)
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% coprime_0_left_iff
tff(fact_5496_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))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_5497_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))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_5498_is__unit__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5499_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_5500_rotate__id,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
     => ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).

% rotate_id
tff(fact_5501_length__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),zip(A,B,Xs,Ys2)) = 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)),Ys2)) ).

% length_zip
tff(fact_5502_normalize__stable,axiom,
    ! [Q5: int,P2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q5))
     => ( algebr8660921524188924756oprime(int,P2,Q5)
       => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P2),Q5)) = aa(int,product_prod(int,int),product_Pair(int,int,P2),Q5) ) ) ) ).

% normalize_stable
tff(fact_5503_is__unit__right__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_right_imp_coprime
tff(fact_5504_is__unit__left__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_left_imp_coprime
tff(fact_5505_coprime__common__divisor,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A))) ) ) ) ) ).

% coprime_common_divisor
tff(fact_5506_coprime__absorb__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y2),X))
         => ( algebr8660921524188924756oprime(A,X,Y2)
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y2),one_one(A))) ) ) ) ).

% coprime_absorb_right
tff(fact_5507_coprime__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,D2: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,D2)
         => ( ! [E: A] :
                ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E),one_one(A)))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E),B2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E),C2)) ) ) )
           => ( ! [E: A] :
                  ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E),one_one(A)))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E),B2))
                     => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E),D2)) ) ) )
             => algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_5508_coprime__absorb__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y2))
         => ( algebr8660921524188924756oprime(A,X,Y2)
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A))) ) ) ) ).

% coprime_absorb_left
tff(fact_5509_not__coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
             => ~ algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% not_coprimeI
tff(fact_5510_not__coprimeE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ algebr8660921524188924756oprime(A,A3,B2)
         => ~ ! [C3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A))) ) ) ) ) ).

% not_coprimeE
tff(fact_5511_coprime__def,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
        <=> ! [C4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),B2))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),one_one(A))) ) ) ) ) ).

% coprime_def
tff(fact_5512_coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ! [C3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A))) ) )
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprimeI
tff(fact_5513_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_5514_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_5515_coprime__divisors,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),D2))
           => ( algebr8660921524188924756oprime(A,C2,D2)
             => algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% coprime_divisors
tff(fact_5516_coprime__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,A3)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_commute
tff(fact_5517_coprime__1__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).

% coprime_1_left
tff(fact_5518_coprime__1__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).

% coprime_1_right
tff(fact_5519_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_5520_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_5521_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_5522_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_5523_rotate__conv__mod,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,N),Xs) = aa(list(A),list(A),rotate(A,modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% rotate_conv_mod
tff(fact_5524_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_5525_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_5526_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) )
         => ? [A9: A,B9: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A9),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),B9),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & algebr8660921524188924756oprime(A,A9,B9) ) ) ) ).

% gcd_coprime_exists
tff(fact_5527_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_5528_Rat__cases,axiom,
    ! [Q5: rat] :
      ~ ! [A6: int,B4: int] :
          ( ( Q5 = aa(int,rat,aa(int,fun(int,rat),fract,A6),B4) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
           => ~ algebr8660921524188924756oprime(int,A6,B4) ) ) ).

% Rat_cases
tff(fact_5529_Rat__induct,axiom,
    ! [P: fun(rat,bool),Q5: rat] :
      ( ! [A6: int,B4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
         => ( algebr8660921524188924756oprime(int,A6,B4)
           => pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A6),B4))) ) )
     => pp(aa(rat,bool,P,Q5)) ) ).

% Rat_induct
tff(fact_5530_coprime__common__divisor__int,axiom,
    ! [A3: int,B2: int,X: int] :
      ( algebr8660921524188924756oprime(int,A3,B2)
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),A3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),B2))
         => ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_5531_zip__obtain__same__length,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),P: fun(list(product_prod(A,B)),bool)] :
      ( ! [Zs2: list(A),Ws: list(B),N2: nat] :
          ( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(B),nat,size_size(list(B)),Ws) )
         => ( ( N2 = 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)),Ys2)) )
           => ( ( Zs2 = take(A,N2,Xs) )
             => ( ( Ws = take(B,N2,Ys2) )
               => pp(aa(list(product_prod(A,B)),bool,P,zip(A,B,Zs2,Ws))) ) ) ) )
     => pp(aa(list(product_prod(A,B)),bool,P,zip(A,B,Xs,Ys2))) ) ).

% zip_obtain_same_length
tff(fact_5532_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),Y2: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y2),aa(list(B),set(B),set2(B),Ys2)))
       => ~ ! [X2: A] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X2),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_5533_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ~ ! [Y3: B] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X),Y3)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_5534_Rat__cases__nonzero,axiom,
    ! [Q5: rat] :
      ( ! [A6: int,B4: int] :
          ( ( Q5 = aa(int,rat,aa(int,fun(int,rat),fract,A6),B4) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
           => ( ( A6 != zero_zero(int) )
             => ~ algebr8660921524188924756oprime(int,A6,B4) ) ) )
     => ( Q5 = zero_zero(rat) ) ) ).

% Rat_cases_nonzero
tff(fact_5535_quotient__of__unique,axiom,
    ! [R: rat] :
    ? [X2: product_prod(int,int)] :
      ( ( R = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),X2)),aa(product_prod(int,int),int,product_snd(int,int),X2)) )
      & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),X2)))
      & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),X2),aa(product_prod(int,int),int,product_snd(int,int),X2))
      & ! [Y: product_prod(int,int)] :
          ( ( ( R = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Y)),aa(product_prod(int,int),int,product_snd(int,int),Y)) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Y)))
            & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y),aa(product_prod(int,int),int,product_snd(int,int),Y)) )
         => ( Y = X2 ) ) ) ).

% quotient_of_unique
tff(fact_5536_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),P2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
    <=> ? [N5: nat] :
          ( ( aa(nat,A,nth(A,Xs),N5) = aa(product_prod(A,B),A,product_fst(A,B),P2) )
          & ( aa(nat,B,nth(B,Ys2),N5) = aa(product_prod(A,B),B,product_snd(A,B),P2) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(B),nat,size_size(list(B)),Ys2))) ) ) ).

% in_set_zip
tff(fact_5537_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)) = collect(product_prod(A,B),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_ob(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys2)) ).

% set_zip
tff(fact_5538_Rats__cases_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),field_char_0_Rats(A)))
         => ~ ! [A6: int,B4: int] :
                ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
               => ( algebr8660921524188924756oprime(int,A6,B4)
                 => ( X != aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A6)),aa(int,A,ring_1_of_int(A),B4)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_5539_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),I: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( distinct(A,Xs)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys2)))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),aa(nat,A,nth(A,Xs),I)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),I)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_5540_Rats__abs__iff,axiom,
    ! [X: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,abs_abs(real),X)),field_char_0_Rats(real)))
    <=> pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),field_char_0_Rats(real))) ) ).

% Rats_abs_iff
tff(fact_5541_Rats__minus__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,uminus_uminus(A),A3)),field_char_0_Rats(A)))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field_char_0_Rats(A))) ) ) ).

% Rats_minus_iff
tff(fact_5542_coprime__int__iff,axiom,
    ! [M2: nat,N: nat] :
      ( algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))
    <=> algebr8660921524188924756oprime(nat,M2,N) ) ).

% coprime_int_iff
tff(fact_5543_coprime__nat__abs__left__iff,axiom,
    ! [K: int,N: nat] :
      ( algebr8660921524188924756oprime(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),N)
    <=> algebr8660921524188924756oprime(int,K,aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% coprime_nat_abs_left_iff
tff(fact_5544_coprime__nat__abs__right__iff,axiom,
    ! [N: nat,K: int] :
      ( algebr8660921524188924756oprime(nat,N,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
    <=> algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),N),K) ) ).

% coprime_nat_abs_right_iff
tff(fact_5545_map__of__zip__is__None,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),X) = none(B) )
      <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% map_of_zip_is_None
tff(fact_5546_coprime__common__divisor__nat,axiom,
    ! [A3: nat,B2: nat,X: nat] :
      ( algebr8660921524188924756oprime(nat,A3,B2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),A3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),B2))
         => ( X = one_one(nat) ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_5547_coprime__Suc__0__right,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_5548_coprime__Suc__0__left,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),N) ).

% coprime_Suc_0_left
tff(fact_5549_coprime__Suc__right__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,N)) ).

% coprime_Suc_right_nat
tff(fact_5550_coprime__Suc__left__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,N),N) ).

% coprime_Suc_left_nat
tff(fact_5551_Rats__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),field_char_0_Rats(A))) ) ).

% Rats_0
tff(fact_5552_Rats__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),field_char_0_Rats(A))) ) ).

% Rats_1
tff(fact_5553_exE__some,axiom,
    ! [A: $tType,P: fun(A,bool),C2: A] :
      ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
     => ( ( C2 = fChoice(A,P) )
       => pp(aa(A,bool,P,C2)) ) ) ).

% exE_some
tff(fact_5554_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X2: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),field_char_0_Rats(real)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),X2)) ) ).

% Rats_no_top_le
tff(fact_5555_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X2: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),field_char_0_Rats(real)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),X)) ) ).

% Rats_no_bot_less
tff(fact_5556_Rats__dense__in__real,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
     => ? [X2: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),field_char_0_Rats(real)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),X2))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),Y2)) ) ) ).

% Rats_dense_in_real
tff(fact_5557_Rats__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field_char_0_Rats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field_char_0_Rats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_divide
tff(fact_5558_Rats__diff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field_char_0_Rats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field_char_0_Rats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_diff
tff(fact_5559_Rats__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,semiring_1_of_nat(A),N)),field_char_0_Rats(A))) ) ).

% Rats_of_nat
tff(fact_5560_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),field_char_0_Rats(real)))
     => ~ ! [M3: nat,N2: nat] :
            ( ( N2 != zero_zero(nat) )
           => ( ( aa(real,real,abs_abs(real),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),M3)),aa(nat,real,semiring_1_of_nat(real),N2)) )
             => ~ algebr8660921524188924756oprime(nat,M3,N2) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_5561_map__of__zip__inject,axiom,
    ! [B: $tType,A: $tType,Ys2: list(A),Xs: list(B),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys2) = 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,Ys2)) = map_of(B,A,zip(B,A,Xs,Zs)) )
           => ( Ys2 = Zs ) ) ) ) ) ).

% map_of_zip_inject
tff(fact_5562_map__of__eq__None__iff,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),X: B] :
      ( ( aa(B,option(A),map_of(B,A,Xys),X) = none(A) )
    <=> ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),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_5563_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_5564_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),N) ) ).

% coprime_diff_one_left_nat
tff(fact_5565_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = collect(A,aTP_Lamp_oc(list(A),fun(A,bool),Xs)) ).

% set_conv_nth
tff(fact_5566_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
      <=> ? [Y4: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),X) = aa(B,option(B),some(B),Y4) ) ) ).

% map_of_zip_is_Some
tff(fact_5567_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( ( Deg2 = Xa )
                & ! [X2: vEBT_VEBT] :
                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                   => vEBT_VEBT_valid(X2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_od(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5568_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_valid(X,Xa)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ~ ( ( Deg2 = Xa )
                  & ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                     => vEBT_VEBT_valid(X3,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                  & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                  & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                  & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_od(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_5569_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: bool] :
      ( ( vEBT_VEBT_valid(X,Xa)
      <=> pp(Y2) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y2)
          <=> ( Xa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( pp(Y2)
              <=> ~ ( ( Deg2 = Xa )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_VEBT_valid(X4,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                    & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_od(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5570_Rats__eq__int__div__int,axiom,
    field_char_0_Rats(real) = collect(real,aTP_Lamp_oe(real,bool)) ).

% Rats_eq_int_div_int
tff(fact_5571_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = collect(real,aTP_Lamp_of(real,bool)) ).

% Rats_eq_int_div_nat
tff(fact_5572_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg4: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary),Deg4)
    <=> ( ( Deg = Deg4 )
        & ! [X4: vEBT_VEBT] :
            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
           => vEBT_VEBT_valid(X4,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_valid(Summary,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),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) = 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),Deg),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
        & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_od(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg),TreeList),Summary))),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5573_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(Uu2,Uv2)),Xa))
               => ( Xa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xa))
                 => ( ( Deg2 = Xa )
                    & ! [X2: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_VEBT_valid(X2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                    & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_od(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_5574_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_valid(X,Xa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(Uu2,Uv2)),Xa))
               => ( Xa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xa))
                 => ~ ( ( Deg2 = Xa )
                      & ! [X3: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_VEBT_valid(X3,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                      & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_od(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_5575_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: bool] :
      ( ( vEBT_VEBT_valid(X,Xa)
      <=> pp(Y2) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y2)
                <=> ( Xa = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(Uu2,Uv2)),Xa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( ( pp(Y2)
                  <=> ( ( Deg2 = Xa )
                      & ! [X4: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_VEBT_valid(X4,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                      & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_od(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2))),Mima)) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_5576_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [X4: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),X4),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys2))))
           => pp(aa(product_prod(A,A),bool,product_case_prod(A,A,bool,fequal(A)),X4)) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_5577_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys2: list(list(A))] :
      ( ! [X2: product_prod(list(A),list(A))] :
          ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),X2),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,Ys2))))
         => pp(aa(product_prod(list(A),list(A)),bool,product_case_prod(list(A),list(A),bool,aTP_Lamp_og(list(A),fun(list(A),bool))),X2)) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys2) )
       => ( ( concat(A,Xs) = concat(A,Ys2) )
        <=> ( Xs = Ys2 ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_5578_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys2: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys2) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys2) )
       => ( ! [X2: product_prod(list(A),list(A))] :
              ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),X2),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,Ys2))))
             => pp(aa(product_prod(list(A),list(A)),bool,product_case_prod(list(A),list(A),bool,aTP_Lamp_og(list(A),fun(list(A),bool))),X2)) )
         => ( Xs = Ys2 ) ) ) ) ).

% concat_injective
tff(fact_5579_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys2)),listrel(A,B,R)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
           => pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,aTP_Lamp_oh(set(product_prod(A,B)),fun(A,fun(B,bool)),R)),X4)) ) ) ) ).

% listrel_iff_zip
tff(fact_5580_ran__map__of__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( distinct(A,Xs)
       => ( ran(A,B,map_of(A,B,zip(A,B,Xs,Ys2))) = aa(list(B),set(B),set2(B),Ys2) ) ) ) ).

% ran_map_of_zip
tff(fact_5581_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_oi(A,bool)) ) ) ).

% Nats_altdef1
tff(fact_5582_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_oj(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_5583_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),semiring_1_Nats(A)))
         => ~ ! [N2: nat] : X != aa(nat,A,semiring_1_of_nat(A),N2) ) ) ).

% Nats_cases
tff(fact_5584_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A,P: fun(A,bool)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),semiring_1_Nats(A)))
         => ( ! [N2: nat] : pp(aa(A,bool,P,aa(nat,A,semiring_1_of_nat(A),N2)))
           => pp(aa(A,bool,P,X)) ) ) ) ).

% Nats_induct
tff(fact_5585_of__nat__in__Nats,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_1_Nats(A))) ) ).

% of_nat_in_Nats
tff(fact_5586_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_mult
tff(fact_5587_Nats__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [W: num] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(num,A,numeral_numeral(A),W)),semiring_1_Nats(A))) ) ).

% Nats_numeral
tff(fact_5588_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_add
tff(fact_5589_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),semiring_1_Nats(A))) ) ).

% Nats_0
tff(fact_5590_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),semiring_1_Nats(A))) ) ).

% Nats_1
tff(fact_5591_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys2)),listrel(A,B,R)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) ) ) ).

% listrel_eq_len
tff(fact_5592_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),semiring_1_Nats(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ) ).

% Nats_diff
tff(fact_5593_Nats__subset__Ints,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),semiring_1_Nats(A)),ring_1_Ints(A))) ) ).

% Nats_subset_Ints
tff(fact_5594_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_ok(A,bool)) ) ) ).

% Nats_altdef2
tff(fact_5595_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys2)),listrel(A,B,R)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),N5)),aa(nat,B,nth(B,Ys2),N5))),R)) ) ) ) ).

% listrel_iff_nth
tff(fact_5596_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I5)) = collect(A,aa(set(nat),fun(A,bool),aTP_Lamp_ol(list(A),fun(set(nat),fun(A,bool)),Xs),I5)) ).

% set_nths
tff(fact_5597_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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,X)) = 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)),X)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_5598_Rat_Opositive_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),bool,bool,ratrel,fequal(bool)),aTP_Lamp_ns(product_prod(int,int),bool)),aTP_Lamp_ns(product_prod(int,int),bool))) ).

% Rat.positive.rsp
tff(fact_5599_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( aa(list(A),A,groups8242544230860333062m_list(A),Ns) = zero_zero(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ns)))
             => ( X4 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_5600_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),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)))
             => ( pp(aa(fun(B,B),bool,aa(fun(A,A),fun(fun(B,B),bool),bNF_rel_fun(A,B,A,B,R2,R2),uminus_uminus(A)),uminus_uminus(B)))
               => pp(aa(fun(int,B),bool,aa(fun(int,A),fun(fun(int,B),bool),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_5601_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,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),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)))
             => pp(aa(fun(num,B),bool,aa(fun(num,A),fun(fun(num,B),bool),bNF_rel_fun(num,num,A,B,fequal(num),R2),numeral_numeral(A)),numeral_numeral(B))) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_5602_power__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & power(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),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)))
           => pp(aa(fun(B,fun(nat,B)),bool,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),bool),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_5603_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),I3),I5)) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_5604_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A))) ) ) ).

% sum_list_nonpos
tff(fact_5605_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2)) )
         => ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_5606_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_5607_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat,C2: A] : aa(list(A),A,groups8242544230860333062m_list(A),replicate(A,N,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),C2) ) ).

% sum_list_replicate
tff(fact_5608_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns)))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_5609_card__length__sum__list__rec,axiom,
    ! [M2: nat,N6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M2))
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_om(nat,fun(nat,fun(list(nat),bool)),M2),N6))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_on(nat,fun(nat,fun(list(nat),bool)),M2),N6)))),aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_oo(nat,fun(nat,fun(list(nat),bool)),M2),N6)))) ) ) ).

% card_length_sum_list_rec
tff(fact_5610_card__length__sum__list,axiom,
    ! [M2: nat,N6: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_om(nat,fun(nat,fun(list(nat),bool)),M2),N6))) = 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),N6),M2)),one_one(nat))),N6) ).

% card_length_sum_list
tff(fact_5611_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),collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_op(list(A),fun(set(nat),fun(nat,bool)),Xs),I5))) ).

% length_nths
tff(fact_5612_of__rat_Orsp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(fun(product_prod(int,int),A),bool,aa(fun(product_prod(int,int),A),fun(fun(product_prod(int,int),A),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),A,A,ratrel,fequal(A)),aTP_Lamp_nt(product_prod(int,int),A)),aTP_Lamp_nt(product_prod(int,int),A))) ) ).

% of_rat.rsp
tff(fact_5613_sum__list__sum__nth,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),Xs) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),nth(B,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_5614_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),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)))
             => pp(aa(fun(nat,B),bool,aa(fun(nat,A),fun(fun(nat,B),bool),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_5615_vanishes__mult__bounded,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( ? [A10: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A10))
          & ! [N2: nat] : pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N2))),A10)) )
     => ( pp(vanishes(Y5))
       => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y5))) ) ) ).

% vanishes_mult_bounded
tff(fact_5616_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( image(nat,nat,aTP_Lamp_or(nat,fun(nat,nat),N),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_5617_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite(option(A)),top_top(set(option(A)))))
    <=> pp(aa(set(A),bool,finite_finite(A),top_top(set(A)))) ) ).

% finite_option_UNIV
tff(fact_5618_inf__top_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),top_top(A)) = A3 ) ).

% inf_top.right_neutral
tff(fact_5619_inf__top_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B2: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
        <=> ( ( A3 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_5620_inf__top_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),A3) = A3 ) ).

% inf_top.left_neutral
tff(fact_5621_inf__top_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = top_top(A) )
        <=> ( ( A3 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_5622_top__eq__inf__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A,Y2: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) )
        <=> ( ( X = top_top(A) )
            & ( Y2 = top_top(A) ) ) ) ) ).

% top_eq_inf_iff
tff(fact_5623_inf__eq__top__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) = top_top(A) )
        <=> ( ( X = top_top(A) )
            & ( Y2 = top_top(A) ) ) ) ) ).

% inf_eq_top_iff
tff(fact_5624_inf__top__right,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),top_top(A)) = X ) ).

% inf_top_right
tff(fact_5625_inf__top__left,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),X) = X ) ).

% inf_top_left
tff(fact_5626_vanishes__const,axiom,
    ! [C2: rat] :
      ( pp(vanishes(aTP_Lamp_os(rat,fun(nat,rat),C2)))
    <=> ( C2 = zero_zero(rat) ) ) ).

% vanishes_const
tff(fact_5627_range__diff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),A3),top_top(set(A))) = top_top(set(A)) ) ).

% range_diff
tff(fact_5628_boolean__algebra_Ocompl__one,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),top_top(A)) = bot_bot(A) ) ) ).

% boolean_algebra.compl_one
tff(fact_5629_boolean__algebra_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),bot_bot(A)) = top_top(A) ) ) ).

% boolean_algebra.compl_zero
tff(fact_5630_Diff__UNIV,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_5631_surj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),N: 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)),N),F2),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_5632_finite__compl,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),uminus_uminus(set(A)),A4)))
      <=> pp(aa(set(A),bool,finite_finite(A),top_top(set(A)))) ) ) ).

% finite_compl
tff(fact_5633_Gcd__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).

% Gcd_UNIV
tff(fact_5634_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : image(A,A,aTP_Lamp_mj(A,fun(A,A),A3),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_5635_uminus__integer_Orsp,axiom,
    pp(aa(fun(int,int),bool,aa(fun(int,int),fun(fun(int,int),bool),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),uminus_uminus(int)),uminus_uminus(int))) ).

% uminus_integer.rsp
tff(fact_5636_Suc_Orsp,axiom,
    pp(aa(fun(nat,nat),bool,aa(fun(nat,nat),fun(fun(nat,nat),bool),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),suc),suc)) ).

% Suc.rsp
tff(fact_5637_less__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less(nat)),ord_less(nat))) ).

% less_natural.rsp
tff(fact_5638_less__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(int,int,fun(int,bool),fun(int,bool),fequal(int),bNF_rel_fun(int,int,bool,bool,fequal(int),fequal(bool))),ord_less(int)),ord_less(int))) ).

% less_integer.rsp
tff(fact_5639_sub_Orsp,axiom,
    pp(aa(fun(num,fun(num,int)),bool,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,int)),bool),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_ot(num,fun(num,int))),aTP_Lamp_ot(num,fun(num,int)))) ).

% sub.rsp
tff(fact_5640_minus__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),minus_minus(nat)),minus_minus(nat))) ).

% minus_natural.rsp
tff(fact_5641_minus__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),minus_minus(int)),minus_minus(int))) ).

% minus_integer.rsp
tff(fact_5642_divide__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),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_5643_divide__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),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_5644_integer__of__natural_Orsp,axiom,
    pp(aa(fun(nat,int),bool,aa(fun(nat,int),fun(fun(nat,int),bool),bNF_rel_fun(nat,nat,int,int,fequal(nat),fequal(int)),semiring_1_of_nat(int)),semiring_1_of_nat(int))) ).

% integer_of_natural.rsp
tff(fact_5645_num_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),bool,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)))),bool),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_5646_Fract_Orsp,axiom,
    pp(aa(fun(int,fun(int,product_prod(int,int))),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int)),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),product_prod(int,int),fequal(int),ratrel)),aTP_Lamp_ou(int,fun(int,product_prod(int,int)))),aTP_Lamp_ou(int,fun(int,product_prod(int,int))))) ).

% Fract.rsp
tff(fact_5647_vanishes__minus,axiom,
    ! [X6: fun(nat,rat)] :
      ( pp(vanishes(X6))
     => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ov(fun(nat,rat),fun(nat,rat)),X6))) ) ).

% vanishes_minus
tff(fact_5648_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( ( A3 != top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),top_top(A))) ) ) ).

% top.not_eq_extremum
tff(fact_5649_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A3)) ) ).

% top.extremum_strict
tff(fact_5650_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( pp(aa(set(fun(A,B)),bool,finite_finite(fun(A,B)),top_top(set(fun(A,B)))))
     => ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => pp(aa(set(A),bool,finite_finite(A),top_top(set(A)))) ) ) ).

% finite_fun_UNIVD1
tff(fact_5651_vanishes__add,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( pp(vanishes(X6))
     => ( pp(vanishes(Y5))
       => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ow(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y5))) ) ) ).

% vanishes_add
tff(fact_5652_vanishes__diff,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( pp(vanishes(X6))
     => ( pp(vanishes(Y5))
       => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ox(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y5))) ) ) ).

% vanishes_diff
tff(fact_5653_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => pp(aa(fun(bool,B),bool,aa(fun(bool,A),fun(fun(bool,B),bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),R2),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).

% transfer_rule_of_bool
tff(fact_5654_bij__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => bij_betw(A,A,uminus_uminus(A),top_top(set(A)),top_top(set(A))) ) ).

% bij_uminus
tff(fact_5655_Compl__UNIV__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),top_top(set(A))) = bot_bot(set(A)) ).

% Compl_UNIV_eq
tff(fact_5656_Compl__empty__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),bot_bot(set(A))) = top_top(set(A)) ).

% Compl_empty_eq
tff(fact_5657_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A4) ).

% Compl_eq_Diff_UNIV
tff(fact_5658_bij__fn,axiom,
    ! [A: $tType,F2: fun(A,A),N: 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)),N),F2),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_5659_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_5660_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),image(B,A,F2,A4))),image(B,A,F2,aa(set(B),set(B),uminus_uminus(set(B)),A4)))) ) ).

% surj_Compl_image_subset
tff(fact_5661_bij__image__Compl__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( image(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),A4)) = aa(set(B),set(B),uminus_uminus(set(B)),image(A,B,F2,A4)) ) ) ).

% bij_image_Compl_eq
tff(fact_5662_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_5663_finite__range__Some,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite(option(A)),image(A,option(A),some(A),top_top(set(A)))))
    <=> pp(aa(set(A),bool,finite_finite(A),top_top(set(A)))) ) ).

% finite_range_Some
tff(fact_5664_notin__range__Some,axiom,
    ! [A: $tType,X: option(A)] :
      ( ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),X),image(A,option(A),some(A),top_top(set(A)))))
    <=> ( X = none(A) ) ) ).

% notin_range_Some
tff(fact_5665_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_5666_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( pp(aa(set(A),bool,finite_finite(A),top_top(set(A))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ).

% finite_UNIV_card_ge_0
tff(fact_5667_uminus__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_nv(product_prod(int,int),product_prod(int,int))),aTP_Lamp_nv(product_prod(int,int),product_prod(int,int)))) ).

% uminus_rat.rsp
tff(fact_5668_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite(A),image(B,A,F2,top_top(set(B)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),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_5669_vanishes__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( pp(vanishes(X6))
    <=> ! [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
         => ? [K3: nat] :
            ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N5))),R5)) ) ) ) ).

% vanishes_def
tff(fact_5670_vanishesI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3))
         => ? [K4: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K4),N2))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N2))),R3)) ) )
     => pp(vanishes(X6)) ) ).

% vanishesI
tff(fact_5671_vanishesD,axiom,
    ! [X6: fun(nat,rat),R: rat] :
      ( pp(vanishes(X6))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R))
       => ? [K2: nat] :
          ! [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N4))
           => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N4))),R)) ) ) ) ).

% vanishesD
tff(fact_5672_inverse__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_nu(product_prod(int,int),product_prod(int,int))),aTP_Lamp_nu(product_prod(int,int),product_prod(int,int)))) ).

% inverse_rat.rsp
tff(fact_5673_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image(nat,nat,suc,top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_5674_inverse__rat_Otransfer,axiom,
    pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_nu(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat))) ).

% inverse_rat.transfer
tff(fact_5675_Rat_Opositive_Otransfer,axiom,
    pp(aa(fun(rat,bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(rat,bool),bool),bNF_rel_fun(product_prod(int,int),rat,bool,bool,pcr_rat,fequal(bool)),aTP_Lamp_ns(product_prod(int,int),bool)),positive)) ).

% Rat.positive.transfer
tff(fact_5676_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_5677_range__abs__Nats,axiom,
    image(int,int,abs_abs(int),top_top(set(int))) = semiring_1_Nats(int) ).

% range_abs_Nats
tff(fact_5678_infinite__UNIV__int,axiom,
    ~ pp(aa(set(int),bool,finite_finite(int),top_top(set(int)))) ).

% infinite_UNIV_int
tff(fact_5679_surj__prod__encode,axiom,
    image(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat)))) = top_top(set(nat)) ).

% surj_prod_encode
tff(fact_5680_bij__prod__encode,axiom,
    bij_betw(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat))),top_top(set(nat))) ).

% bij_prod_encode
tff(fact_5681_Ints__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_Ints(A) = image(int,A,ring_1_of_int(A),top_top(set(int))) ) ) ).

% Ints_def
tff(fact_5682_int__in__range__abs,axiom,
    ! [N: nat] : pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(nat,int,semiring_1_of_nat(int),N)),image(int,int,abs_abs(int),top_top(set(int))))) ).

% int_in_range_abs
tff(fact_5683_one__rat_Otransfer,axiom,
    pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))),one_one(rat))) ).

% one_rat.transfer
tff(fact_5684_zero__rat_Otransfer,axiom,
    pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int))),zero_zero(rat))) ).

% zero_rat.transfer
tff(fact_5685_Fract_Otransfer,axiom,
    pp(aa(fun(int,fun(int,rat)),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),rat,fequal(int),pcr_rat)),aTP_Lamp_ou(int,fun(int,product_prod(int,int)))),fract)) ).

% Fract.transfer
tff(fact_5686_of__rat_Otransfer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(fun(rat,A),bool,aa(fun(product_prod(int,int),A),fun(fun(rat,A),bool),bNF_rel_fun(product_prod(int,int),rat,A,A,pcr_rat,fequal(A)),aTP_Lamp_nt(product_prod(int,int),A)),field_char_0_of_rat(A))) ) ).

% of_rat.transfer
tff(fact_5687_uminus__rat_Otransfer,axiom,
    pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_nv(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat))) ).

% uminus_rat.transfer
tff(fact_5688_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(real,real,root(N),X) = zero_zero(real) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(real,real,root(N),X) = the_inv_into(real,real,top_top(set(real)),aTP_Lamp_oy(nat,fun(real,real),N),X) ) ) ) ).

% root_def
tff(fact_5689_times__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).

% times_int.transfer
tff(fact_5690_num_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),bool,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)))),bool),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_5691_verit__eq__simplify_I20_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X23: 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,X23)) = aa(A,A,aa(num,fun(A,A),F22,X23),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),X23)) ).

% verit_eq_simplify(20)
tff(fact_5692_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_5693_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_5694_zero__int_Otransfer,axiom,
    pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))),zero_zero(int))) ).

% zero_int.transfer
tff(fact_5695_int__transfer,axiom,
    pp(aa(fun(nat,int),bool,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),bool),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_oz(nat,product_prod(nat,nat))),semiring_1_of_nat(int))) ).

% int_transfer
tff(fact_5696_uminus__int_Otransfer,axiom,
    pp(aa(fun(int,int),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),bool),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_lq(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).

% uminus_int.transfer
tff(fact_5697_nat_Otransfer,axiom,
    pp(aa(fun(int,nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(int,nat),bool),bNF_rel_fun(product_prod(nat,nat),int,nat,nat,pcr_int,fequal(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))),nat2)) ).

% nat.transfer
tff(fact_5698_one__int_Otransfer,axiom,
    pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))),one_one(int))) ).

% one_int.transfer
tff(fact_5699_of__int_Otransfer,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(fun(int,A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(int,A),bool),bNF_rel_fun(product_prod(nat,nat),int,A,A,pcr_int,fequal(A)),product_case_prod(nat,nat,A,aTP_Lamp_lr(nat,fun(nat,A)))),ring_1_of_int(A))) ) ).

% of_int.transfer
tff(fact_5700_less__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lt(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less(int))) ).

% less_int.transfer
tff(fact_5701_less__eq__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lv(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).

% less_eq_int.transfer
tff(fact_5702_plus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).

% plus_int.transfer
tff(fact_5703_minus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).

% minus_int.transfer
tff(fact_5704_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
         => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_5705_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D6: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( X != zero_zero(real) )
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
             => ( D6 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
               => ( D6 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
               => ( D6 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(N),D6,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_5706_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => has_field_derivative(real,aTP_Lamp_pa(real,real),suminf(real,aTP_Lamp_pb(real,fun(nat,real),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_5707_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D5: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
            & ! [H3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H3),D5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H3))),aa(real,real,F2,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_5708_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D5: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
            & ! [H3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H3),D5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H3)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_5709_DERIV__const__ratio__const2,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A3 != B2 )
     => ( ! [X2: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X2,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A3))),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_5710_DERIV__const__ratio__const,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A3 != B2 )
     => ( ! [X2: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X2,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,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)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_5711_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M2: A,X: A] :
          ( has_field_derivative(A,G,M2,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_pc(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),aa(A,A,G,X))),M2),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_5712_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M2: A,X: A] :
          ( has_field_derivative(A,G,M2,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_pd(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),aa(A,A,G,X))),M2),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_5713_DERIV__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : has_field_derivative(A,exp(A),aa(A,A,exp(A),X),topolo174197925503356063within(A,X,top_top(set(A)))) ) ).

% DERIV_exp
tff(fact_5714_DERIV__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : has_field_derivative(A,sin(A),aa(A,A,cos(A),X),topolo174197925503356063within(A,X,top_top(set(A)))) ) ).

% DERIV_sin
tff(fact_5715_DERIV__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : has_field_derivative(A,cos(A),aa(A,A,uminus_uminus(A),aa(A,A,sin(A),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ).

% DERIV_cos
tff(fact_5716_DERIV__mirror,axiom,
    ! [F2: fun(real,real),Y2: real,X: real] :
      ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),X),top_top(set(real))))
    <=> has_field_derivative(real,aTP_Lamp_pe(fun(real,real),fun(real,real),F2),aa(real,real,uminus_uminus(real),Y2),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_mirror
tff(fact_5717_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_pf(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,F2,X))),D6)),aa(A,A,inverse_inverse(A),aa(A,A,F2,X)))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_5718_DERIV__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aTP_Lamp_pg(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),D6),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_minus
tff(fact_5719_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,F6,F4)
         => has_field_derivative(A,aTP_Lamp_pg(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),F6),F4) ) ) ).

% field_differentiable_minus
tff(fact_5720_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F4: filter(A)] : has_field_derivative(A,aTP_Lamp_ph(A,fun(A,A),K),zero_zero(A),F4) ) ).

% DERIV_const
tff(fact_5721_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F4: filter(A)] : has_field_derivative(A,aTP_Lamp_pi(A,A),one_one(A),F4) ) ).

% DERIV_ident
tff(fact_5722_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,F4: filter(A),C2: real] :
          ( has_field_derivative(A,F2,D6,F4)
         => has_field_derivative(A,aa(real,fun(A,A),aTP_Lamp_pj(fun(A,A),fun(real,fun(A,A)),F2),C2),real_V8093663219630862766scaleR(A,C2,D6),F4) ) ) ).

% has_field_derivative_scaleR_right
tff(fact_5723_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_pk(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D6),C2),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_cdivide
tff(fact_5724_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pl(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),D6),E4),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_diff
tff(fact_5725_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,F4: filter(A),G: fun(A,A),G2: A] :
          ( has_field_derivative(A,F2,F6,F4)
         => ( has_field_derivative(A,G,G2,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pl(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),F6),G2),F4) ) ) ) ).

% field_differentiable_diff
tff(fact_5726_has__field__derivative__cosh,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [G: fun(A11,A11),Db: A11,X: A11,S: set(A11)] :
          ( has_field_derivative(A11,G,Db,topolo174197925503356063within(A11,X,S))
         => has_field_derivative(A11,aTP_Lamp_pm(fun(A11,A11),fun(A11,A11),G),aa(A11,A11,aa(A11,fun(A11,A11),times_times(A11),aa(A11,A11,sinh(A11),aa(A11,A11,G,X))),Db),topolo174197925503356063within(A11,X,S)) ) ) ).

% has_field_derivative_cosh
tff(fact_5727_has__field__derivative__sinh,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [G: fun(A11,A11),Db: A11,X: A11,S: set(A11)] :
          ( has_field_derivative(A11,G,Db,topolo174197925503356063within(A11,X,S))
         => has_field_derivative(A11,aTP_Lamp_pn(fun(A11,A11),fun(A11,A11),G),aa(A11,A11,aa(A11,fun(A11,A11),times_times(A11),aa(A11,A11,cosh(A11),aa(A11,A11,G,X))),Db),topolo174197925503356063within(A11,X,S)) ) ) ).

% has_field_derivative_sinh
tff(fact_5728_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D5: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
            & ! [H3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H3))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H3)),S3))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H3),D5))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H3))),aa(real,real,F2,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_5729_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S3))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D5: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
            & ! [H3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H3))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H3)),S3))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H3),D5))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_5730_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(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)),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),D6),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,X)),aa(A,A,G,X))),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_5731_MVT2,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),F6: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( ! [X2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2))
             => has_field_derivative(real,F2,aa(real,real,F6,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
       => ? [Z3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Z3))
            & pp(aa(real,bool,aa(real,fun(real,bool),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_5732_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ( ! [Y3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y3))),D2))
             => ( aa(real,real,F2,X) = aa(real,real,F2,Y3) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_5733_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_5734_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M2: A,X: A] :
          ( has_field_derivative(A,G,M2,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_pp(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(A,A,sin(A),aa(A,A,G,X)))),M2),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_5735_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xa: A] : has_field_derivative(A,aTP_Lamp_pq(A,fun(A,A),K),aa(A,A,uminus_uminus(A),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),K))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_5736_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A),N: nat] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_pr(fun(A,A),fun(nat,fun(A,A)),F2),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),N))),aa(A,A,aa(A,fun(A,A),times_times(A),D6),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,X)),N))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_power_Suc
tff(fact_5737_DERIV__const__average,axiom,
    ! [A3: real,B2: real,V2: fun(real,real),K: real] :
      ( ( A3 != B2 )
     => ( ! [X2: real] : has_field_derivative(real,V2,K,topolo174197925503356063within(real,X2,top_top(set(real))))
       => ( aa(real,real,V2,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,V2,A3)),aa(real,real,V2,B2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ) ).

% DERIV_const_average
tff(fact_5738_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,S: set(A)] :
          ( ( X != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_inverse
tff(fact_5739_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A),N: nat] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_ps(fun(A,A),fun(nat,fun(A,A)),F2),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),times_times(A),D6),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_power
tff(fact_5740_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ( ! [Y3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y3))),D2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F2,Y3)),aa(real,real,F2,X))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_5741_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ( ! [Y3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y3))),D2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F2,X)),aa(real,real,F2,Y3))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_5742_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_5743_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_pt(nat,fun(real,real),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,X,S)) ).

% DERIV_pow
tff(fact_5744_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => has_field_derivative(A,aTP_Lamp_pu(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_5745_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
           => ( topolo174197925503356063within(A,X,set_or1337092689740270186AtMost(A,A3,B2)) = topolo174197925503356063within(A,X,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_5746_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M2: real,X: real,N: nat] :
      ( has_field_derivative(real,G,M2,topolo174197925503356063within(real,X,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_pv(fun(real,real),fun(nat,fun(real,real)),G),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,G,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))),M2),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_5747_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5748_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S: set(A),G: fun(A,A),E2: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E2,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(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)),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,X))),aa(A,A,aa(A,fun(A,A),times_times(A),E2),aa(A,A,F2,X)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_5749_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_pf(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,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_5750_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),F2: fun(A,A),F6: A,Z: A] :
          ( ! [Z3: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K6))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_ey(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))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K6))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),C2),Z),F6) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_5751_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Z))
     => has_field_derivative(real,aTP_Lamp_pw(real,fun(real,real),R),aa(real,real,aa(real,fun(real,real),times_times(real),R),aa(real,real,aa(real,fun(real,real),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_5752_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_px(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6)))
               => has_field_derivative(A,aTP_Lamp_pu(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_5753_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6)))
           => has_field_derivative(A,aTP_Lamp_pu(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_5754_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),Z: A] :
          ( ! [Z3: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K6))
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K6))
           => has_field_derivative(A,aTP_Lamp_pu(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_5755_DERIV__log,axiom,
    ! [X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,log2(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_5756_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M2: real,X: real,R: real] :
      ( has_field_derivative(real,G,M2,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,G,X)))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_py(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),aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,G,X)),aa(real,real,aa(real,fun(real,real),minus_minus(real),R),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M2),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_5757_DERIV__powr,axiom,
    ! [G: fun(real,real),M2: real,X: real,F2: fun(real,real),R: real] :
      ( has_field_derivative(real,G,M2,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,G,X)))
       => ( has_field_derivative(real,F2,R,topolo174197925503356063within(real,X,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_pz(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,G,X)),aa(real,real,F2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),aa(real,real,ln_ln(real),aa(real,real,G,X)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),M2),aa(real,real,F2,X))),aa(real,real,G,X)))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_5758_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,cos(A),X) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_5759_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_5760_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F6: fun(real,fun(nat,real)),X0: real,A3: real,B2: real,L4: fun(nat,real)] :
      ( ! [N2: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_qa(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F2),N2),aa(nat,real,aa(real,fun(nat,real),F6,X0),N2),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X2: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),set_or5935395276787703475ssThan(real,A3,B2)))
           => summable(real,aa(real,fun(nat,real),F2,X2)) )
       => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,A3,B2)))
         => ( summable(real,aa(real,fun(nat,real),F6,X0))
           => ( summable(real,L4)
             => ( ! [N2: nat,X2: real,Y3: real] :
                    ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),set_or5935395276787703475ssThan(real,A3,B2)))
                   => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y3),set_or5935395276787703475ssThan(real,A3,B2)))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),F2,X2),N2)),aa(nat,real,aa(real,fun(nat,real),F2,Y3),N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L4,N2)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X2),Y3))))) ) )
               => has_field_derivative(real,aTP_Lamp_qb(fun(real,fun(nat,real)),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),F6,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_5761_DERIV__arctan,axiom,
    ! [X: real] : has_field_derivative(real,arctan,aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(real,X,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_5762_arsinh__real__has__field__derivative,axiom,
    ! [X: 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),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,X,A4)) ).

% arsinh_real_has_field_derivative
tff(fact_5763_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,sin(A),X) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_5764_has__field__derivative__tanh,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [G: fun(A11,A11),X: A11,Db: A11,S: set(A11)] :
          ( ( aa(A11,A11,cosh(A11),aa(A11,A11,G,X)) != zero_zero(A11) )
         => ( has_field_derivative(A11,G,Db,topolo174197925503356063within(A11,X,S))
           => has_field_derivative(A11,aTP_Lamp_qc(fun(A11,A11),fun(A11,A11),G),aa(A11,A11,aa(A11,fun(A11,A11),times_times(A11),aa(A11,A11,aa(A11,fun(A11,A11),minus_minus(A11),one_one(A11)),aa(nat,A11,aa(A11,fun(nat,A11),power_power(A11),aa(A11,A11,tanh(A11),aa(A11,A11,G,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Db),topolo174197925503356063within(A11,X,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_5765_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D6: real] :
      ( ( X != zero_zero(real) )
     => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( D6 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
       => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
           => ( D6 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
         => has_field_derivative(real,sqrt,D6,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_5766_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A4: set(real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => has_field_derivative(real,arcosh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,X,A4)) ) ).

% arcosh_real_has_field_derivative
tff(fact_5767_artanh__real__has__field__derivative,axiom,
    ! [X: real,A4: set(real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => has_field_derivative(real,artanh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(real,X,A4)) ) ).

% artanh_real_has_field_derivative
tff(fact_5768_DERIV__power__series_H,axiom,
    ! [R2: real,F2: fun(nat,real),X0: real] :
      ( ! [X2: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2)))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_qd(fun(nat,real),fun(real,fun(nat,real)),F2),X2)) )
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
         => has_field_derivative(real,aTP_Lamp_qf(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_qd(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_5769_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_5770_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_5771_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_5772_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,N: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M3: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X2),topolo174197925503356063within(real,X2,top_top(set(real)))) )
     => ? [T3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),X)))
          & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_qg(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_5773_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,N: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M3: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X2),topolo174197925503356063within(real,X2,top_top(set(real))))
       => ? [T3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_qg(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_5774_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( ( X != zero_zero(real) )
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_5775_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H),zero_zero(real)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M3: nat,T3: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),H),T3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T3),zero_zero(real))) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
           => ? [T3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H),T3))
                & pp(aa(real,bool,aa(real,fun(real,bool),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,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qh(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_5776_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M3: nat,T3: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T3))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T3),H)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
         => ? [T3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T3))
              & pp(aa(real,bool,aa(real,fun(real,bool),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,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qh(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_5777_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M3: nat,T3: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T3),H)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
           => ? [T3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T3))
                & pp(aa(real,bool,aa(real,fun(real,bool),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,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qh(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_5778_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( X != zero_zero(real) )
         => ( ! [M3: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X2),topolo174197925503356063within(real,X2,top_top(set(real))))
           => ? [T3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T3)))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),X)))
                & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_qg(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_5779_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M3: nat,T3: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),X))) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
       => ? [T3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T3)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_qg(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_5780_Taylor,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B2: real,C2: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M3: nat,T3: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),T3))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T3),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),B2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),B2))
                 => ( ( X != C2 )
                   => ? [T3: real] :
                        ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),C2))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),T3))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T3),C2)) ) )
                        & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),C2))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),T3))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T3),X)) ) )
                        & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qi(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),C2)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_5781_Taylor__up,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M3: nat,T3: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),T3))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T3),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),B2))
             => ? [T3: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),T3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),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,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qj(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_5782_Taylor__down,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M3: nat,T3: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),T3))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T3),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),B2))
             => ? [T3: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),T3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),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,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qj(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A3),C2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T3)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_5783_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B5: real] :
      ( ! [M3: nat,T3: real] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T3),H)) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T3),topolo174197925503356063within(real,T3,top_top(set(real)))) )
     => ( ( N = aa(nat,nat,suc,K) )
       => ! [M: nat,T7: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T7))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),H)) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_ql(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),N),Diff),B5),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)),T7)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_qm(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M),T7)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M))))),aa(real,real,aa(real,fun(real,real),times_times(real),B5),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),T7),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M)))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M))))))),topolo174197925503356063within(real,T7,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_5784_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,G,X)),one_one(real)))
           => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_qn(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_qo(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_5785_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,G,X)),one_one(real)))
           => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_qp(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_qq(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_5786_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( ( aa(real,real,cos(real),aa(A,real,G,X)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_qr(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_qs(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_tan
tff(fact_5787_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F4: filter(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,F4)
         => ( has_derivative(A,B,G,G2,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_qt(fun(A,B),fun(fun(A,B),fun(A,B)),F6),G2),F4) ) ) ) ).

% has_derivative_diff
tff(fact_5788_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F4: filter(A)] : has_derivative(A,B,aTP_Lamp_qu(B,fun(A,B),C2),aTP_Lamp_qv(A,B),F4) ) ).

% has_derivative_const
tff(fact_5789_has__derivative__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F6,F4)
         => has_derivative(A,B,aTP_Lamp_qw(fun(A,B),fun(A,B),F2),aTP_Lamp_qw(fun(A,B),fun(A,B),F6),F4) ) ) ).

% has_derivative_minus
tff(fact_5790_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F4: fun(A,B),X: A] :
          ( has_derivative(A,B,aTP_Lamp_qv(A,B),F4,topolo174197925503356063within(A,X,top_top(set(A))))
         => ! [X3: A] : aa(A,B,F4,X3) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_5791_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_qx(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_qy(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_exp
tff(fact_5792_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_qz(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ra(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_sin
tff(fact_5793_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S))
         => has_derivative(A,A,aTP_Lamp_rb(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),aa(A,A,sinh(A),aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_cosh
tff(fact_5794_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S))
         => has_derivative(A,A,aTP_Lamp_rc(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),aa(A,A,cosh(A),aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_sinh
tff(fact_5795_has__derivative__divide_H,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),F6: fun(C,A),X: C,S3: set(C),G: fun(C,A),G2: fun(C,A)] :
          ( has_derivative(C,A,F2,F6,topolo174197925503356063within(C,X,S3))
         => ( has_derivative(C,A,G,G2,topolo174197925503356063within(C,X,S3))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_rd(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_re(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F2),F6),X),G),G2),topolo174197925503356063within(C,X,S3)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_5796_has__derivative__inverse,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(C,A),X: C,F6: fun(C,A),S3: set(C)] :
          ( ( aa(C,A,F2,X) != zero_zero(A) )
         => ( has_derivative(C,A,F2,F6,topolo174197925503356063within(C,X,S3))
           => has_derivative(C,A,aTP_Lamp_rf(fun(C,A),fun(C,A),F2),aa(fun(C,A),fun(C,A),aa(C,fun(fun(C,A),fun(C,A)),aTP_Lamp_rg(fun(C,A),fun(C,fun(fun(C,A),fun(C,A))),F2),X),F6),topolo174197925503356063within(C,X,S3)) ) ) ) ).

% has_derivative_inverse
tff(fact_5797_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,S3: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_rh(A,fun(A,A),X),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_inverse'
tff(fact_5798_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_ri(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_rj(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_cos
tff(fact_5799_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S3: set(A),N: nat] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S3))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_rk(fun(A,B),fun(nat,fun(A,B)),F2),N),aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_rl(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F6),X),N),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_power
tff(fact_5800_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X)))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_rm(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_rn(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_ln
tff(fact_5801_has__derivative__divide,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(C,A),F6: fun(C,A),X: C,S3: set(C),G: fun(C,A),G2: fun(C,A)] :
          ( has_derivative(C,A,F2,F6,topolo174197925503356063within(C,X,S3))
         => ( has_derivative(C,A,G,G2,topolo174197925503356063within(C,X,S3))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ro(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_rp(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F2),F6),X),G),G2),topolo174197925503356063within(C,X,S3)) ) ) ) ) ).

% has_derivative_divide
tff(fact_5802_has__derivative__prod,axiom,
    ! [B: $tType,I6: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [I5: set(I6),F2: fun(I6,fun(A,B)),F6: fun(I6,fun(A,B)),X: A,S3: set(A)] :
          ( ! [I3: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I3),I5))
             => has_derivative(A,B,aa(I6,fun(A,B),F2,I3),aa(I6,fun(A,B),F6,I3),topolo174197925503356063within(A,X,S3)) )
         => has_derivative(A,B,aa(fun(I6,fun(A,B)),fun(A,B),aTP_Lamp_rr(set(I6),fun(fun(I6,fun(A,B)),fun(A,B)),I5),F2),aa(A,fun(A,B),aa(fun(I6,fun(A,B)),fun(A,fun(A,B)),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B))),aTP_Lamp_rt(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B)))),I5),F2),F6),X),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_prod
tff(fact_5803_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,X6: set(A),F2: fun(A,real),F6: fun(A,real)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,X6))
         => ( has_derivative(A,real,F2,F6,topolo174197925503356063within(A,X,X6))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X)))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ru(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_rv(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G2),X),F2),F6),topolo174197925503356063within(A,X,X6)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_5804_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X)))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_rw(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_rx(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_5805_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_ry(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_rz(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_arctan
tff(fact_5806_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_px(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6)))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sb(fun(nat,A),fun(A,fun(A,A)),C2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_5807_surj__int__encode,axiom,
    image(int,nat,nat_int_encode,top_top(set(int))) = top_top(set(nat)) ).

% surj_int_encode
tff(fact_5808_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6)))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_pu(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_5809_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_sc(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_5810_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_sd(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_5811_power__tendsto__0__iff,axiom,
    ! [A: $tType,N: nat,F2: fun(A,real),F4: filter(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_se(nat,fun(fun(A,real),fun(A,real)),N),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_5812_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))))
           => ( ? [D3: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
                  & ! [X2: A] :
                      ( ( ( X2 != A3 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A3))),D3)) )
                     => ( aa(A,B,F2,X2) != aa(A,B,F2,A3) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_sf(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_5813_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L4: B,A3: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sg(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_5814_LIM__not__zero,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topolo8386298272705272623_space(A)
        & zero(Aa)
        & topological_t2_space(Aa) )
     => ! [K: Aa,A3: A] :
          ( ( K != zero_zero(Aa) )
         => ~ filterlim(A,Aa,aTP_Lamp_sh(Aa,fun(A,Aa),K),topolo7230453075368039082e_nhds(Aa,zero_zero(Aa)),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_not_zero
tff(fact_5815_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [X: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_si(A,fun(fun(A,B),fun(A,B)),X),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,X)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_5816_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_sj(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_5817_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L4: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sj(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_5818_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A,L4: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sj(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_5819_continuous__within__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A,S: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z,S),cos(A)) ) ).

% continuous_within_cos
tff(fact_5820_continuous__within__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A,S: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z,S),sin(A)) ) ).

% continuous_within_sin
tff(fact_5821_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S3))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sk(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_field_derivative_iff
tff(fact_5822_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S3))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sk(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_field_derivativeD
tff(fact_5823_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A3: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( aa(A,A,cos(A),A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_sl(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A3)),F4) ) ) ) ).

% tendsto_tan
tff(fact_5824_tendsto__add__zero,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(D,B),F4: filter(D),G: fun(D,B)] :
          ( filterlim(D,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_sm(fun(D,B),fun(fun(D,B),fun(D,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_5825_tendsto__minus__cancel__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Y2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Y2)),F4)
        <=> filterlim(A,B,aTP_Lamp_sn(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,Y2),F4) ) ) ).

% tendsto_minus_cancel_left
tff(fact_5826_tendsto__minus__cancel,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [F2: fun(B,A),A3: A,F4: filter(B)] :
          ( filterlim(B,A,aTP_Lamp_so(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,uminus_uminus(A),A3)),F4)
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4) ) ) ).

% tendsto_minus_cancel
tff(fact_5827_tendsto__minus,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [F2: fun(B,A),A3: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => filterlim(B,A,aTP_Lamp_so(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,uminus_uminus(A),A3)),F4) ) ) ).

% tendsto_minus
tff(fact_5828_tendsto__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A3: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( A3 != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_sp(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,inverse_inverse(A),A3)),F4) ) ) ) ).

% tendsto_inverse
tff(fact_5829_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_sq(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_5830_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_sq(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_iff
tff(fact_5831_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,real,aTP_Lamp_sq(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_norm_zero
tff(fact_5832_tendsto__ln,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( ( A3 != zero_zero(real) )
       => filterlim(A,real,aTP_Lamp_js(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,ln_ln(real),A3)),F4) ) ) ).

% tendsto_ln
tff(fact_5833_tendsto__powr,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) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,aa(real,fun(real,real),powr(real),A3),B2)),F4) ) ) ) ).

% tendsto_powr
tff(fact_5834_tendsto__sgn,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),L: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
         => ( ( L != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_ss(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,sgn_sgn(A),L)),F4) ) ) ) ).

% tendsto_sgn
tff(fact_5835_continuous__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_st(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_minus
tff(fact_5836_tendsto__uminus__nhds,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [A3: A] : filterlim(A,A,uminus_uminus(A),topolo7230453075368039082e_nhds(A,aa(A,A,uminus_uminus(A),A3)),topolo7230453075368039082e_nhds(A,A3)) ) ).

% tendsto_uminus_nhds
tff(fact_5837_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A3: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( aa(A,A,sin(A),A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_su(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A3)),F4) ) ) ) ).

% tendsto_cot
tff(fact_5838_int__encode__eq,axiom,
    ! [X: int,Y2: int] :
      ( ( aa(int,nat,nat_int_encode,X) = aa(int,nat,nat_int_encode,Y2) )
    <=> ( X = Y2 ) ) ).

% int_encode_eq
tff(fact_5839_continuous__sinh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F4,F2)
         => topolo3448309680560233919inuous(C,A,F4,aTP_Lamp_sv(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_sinh
tff(fact_5840_continuous__cosh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F4,F2)
         => topolo3448309680560233919inuous(C,A,F4,aTP_Lamp_sw(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_cosh
tff(fact_5841_continuous__exp,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F4,F2)
         => topolo3448309680560233919inuous(C,A,F4,aTP_Lamp_sx(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_exp
tff(fact_5842_continuous__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_sy(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cos
tff(fact_5843_tendsto__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A3: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F4)
         => filterlim(A,B,aTP_Lamp_sy(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,cos(B),A3)),F4) ) ) ).

% tendsto_cos
tff(fact_5844_tendsto__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A3: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F4)
         => filterlim(A,B,aTP_Lamp_sz(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,sin(B),A3)),F4) ) ) ).

% tendsto_sin
tff(fact_5845_tendsto__arctan,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F4)
     => filterlim(A,real,aTP_Lamp_ta(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arctan,X)),F4) ) ).

% tendsto_arctan
tff(fact_5846_tendsto__arsinh,axiom,
    ! [B: $tType,F2: fun(B,real),A3: real,F4: filter(B)] :
      ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => filterlim(B,real,aTP_Lamp_tb(fun(B,real),fun(B,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arsinh(real),A3)),F4) ) ).

% tendsto_arsinh
tff(fact_5847_tendsto__cosh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A3: A,F4: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => filterlim(C,A,aTP_Lamp_tc(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cosh(A),A3)),F4) ) ) ).

% tendsto_cosh
tff(fact_5848_tendsto__exp,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A3: A,F4: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => filterlim(C,A,aTP_Lamp_td(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,exp(A),A3)),F4) ) ) ).

% tendsto_exp
tff(fact_5849_tendsto__sinh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A3: A,F4: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => filterlim(C,A,aTP_Lamp_te(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,sinh(A),A3)),F4) ) ) ).

% tendsto_sinh
tff(fact_5850_continuous__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_sz(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sin
tff(fact_5851_continuous__arctan,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_tf(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arctan
tff(fact_5852_continuous__arsinh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_tg(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arsinh
tff(fact_5853_tendsto__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A3: A,F4: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( aa(A,A,cosh(A),A3) != zero_zero(A) )
           => filterlim(C,A,aTP_Lamp_th(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tanh(A),A3)),F4) ) ) ) ).

% tendsto_tanh
tff(fact_5854_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_ti(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_diff
tff(fact_5855_tendsto__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [F2: fun(B,A),A3: A,F4: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F4)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tj(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),F4) ) ) ) ).

% tendsto_diff
tff(fact_5856_tendsto__mult__one,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(D,B),F4: filter(D),G: fun(D,B)] :
          ( filterlim(D,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
         => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
           => filterlim(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_tk(fun(D,B),fun(fun(D,B),fun(D,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F4) ) ) ) ).

% tendsto_mult_one
tff(fact_5857_tendsto__divide__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),F4: filter(B),C2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_tl(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_5858_tendsto__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),A3: A,F4: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F4)
           => ( ( B2 != zero_zero(A) )
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tm(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_5859_tendsto__mult__right__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D,A),F4: filter(D),C2: A] :
          ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => filterlim(D,A,aa(A,fun(D,A),aTP_Lamp_tn(fun(D,A),fun(A,fun(D,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_5860_tendsto__mult__left__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D,A),F4: filter(D),C2: A] :
          ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => filterlim(D,A,aa(A,fun(D,A),aTP_Lamp_to(fun(D,A),fun(A,fun(D,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_5861_tendsto__mult__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D,A),F4: filter(D),G: fun(D,A)] :
          ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => ( filterlim(D,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_tp(fun(D,A),fun(fun(D,A),fun(D,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_5862_Lim__transform__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),G: fun(B,A),F4: filter(B),A3: A] :
          ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tq(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
          <=> filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A3),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_5863_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tr(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_5864_Lim__transform2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A3: A,F4: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tq(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A3),F4) ) ) ) ).

% Lim_transform2
tff(fact_5865_Lim__transform,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(B,A),A3: A,F4: filter(B),F2: fun(B,A)] :
          ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ts(fun(B,A),fun(fun(B,A),fun(B,A)),G),F2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4) ) ) ) ).

% Lim_transform
tff(fact_5866_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tr(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_5867_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tr(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_5868_tendsto__arcosh,axiom,
    ! [B: $tType,F2: fun(B,real),A3: real,F4: filter(B)] :
      ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A3))
       => filterlim(B,real,aTP_Lamp_tt(fun(B,real),fun(B,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A3)),F4) ) ) ).

% tendsto_arcosh
tff(fact_5869_tendsto__null__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(B),F2: fun(A,fun(B,C)),F4: filter(A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
             => filterlim(A,C,aa(B,fun(A,C),aTP_Lamp_tu(fun(A,fun(B,C)),fun(B,fun(A,C)),F2),I3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) )
         => filterlim(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_tv(set(B),fun(fun(A,fun(B,C)),fun(A,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ).

% tendsto_null_sum
tff(fact_5870_tendsto__one__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(B),F2: fun(A,fun(B,C)),F4: filter(A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
             => filterlim(A,C,aa(B,fun(A,C),aTP_Lamp_tw(fun(A,fun(B,C)),fun(B,fun(A,C)),F2),I3),topolo7230453075368039082e_nhds(C,one_one(C)),F4) )
         => filterlim(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_tx(set(B),fun(fun(A,fun(B,C)),fun(A,C)),I5),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F4) ) ) ).

% tendsto_one_prod'
tff(fact_5871_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F4: filter(A),N: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_ty(fun(A,B),fun(nat,fun(A,B)),F2),N),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_5872_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)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
         => ( ( A3 != one_one(real) )
           => ( pp(aa(real,bool,aa(real,fun(real,bool),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,log2(A3),B2)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_5873_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A3))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),one_one(real)))
         => filterlim(A,real,aTP_Lamp_ua(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A3)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_5874_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L: B,A3: A,G: fun(A,C),M2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( ! [X2: A] :
                ( ( X2 != A3 )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(A,C,G,X2)),M2))),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X2)),L)))) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M2),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_5875_LIM__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A3: A,F2: fun(A,D),L4: D] :
          ( nO_MATCH(C,A,zero_zero(C),A3)
         => ( filterlim(A,D,F2,topolo7230453075368039082e_nhds(D,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
          <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_ub(A,fun(fun(A,D),fun(A,D)),A3),F2),topolo7230453075368039082e_nhds(D,L4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_5876_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L4: B,A3: A,R: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
           => ? [S2: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S2))
                & ! [X3: A] :
                    ( ( ( X3 != A3 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A3))),S2)) )
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X3)),L4))),R)) ) ) ) ) ) ).

% LIM_D
tff(fact_5877_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B),L4: B] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [S6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S6))
                  & ! [X2: A] :
                      ( ( ( X2 != A3 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A3))),S6)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X2)),L4))),R3)) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_5878_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L4: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [S5: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S5))
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A3))),S5)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X4)),L4))),R5)) ) ) ) ) ) ).

% LIM_eq
tff(fact_5879_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] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
         => ( ! [X2: A] :
                ( ( X2 != A3 )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A3))),R2))
                 => ( aa(A,B,F2,X2) = aa(A,B,G,X2) ) ) )
           => ( 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_5880_isCont__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),cos(A)) ) ).

% isCont_cos
tff(fact_5881_isCont__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),sin(A)) ) ).

% isCont_sin
tff(fact_5882_isCont__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),exp(A)) ) ).

% isCont_exp
tff(fact_5883_isCont__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),cosh(A)) ) ).

% isCont_cosh
tff(fact_5884_isCont__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),sinh(A)) ) ).

% isCont_sinh
tff(fact_5885_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A3: A,D6: 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,D6),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,D6),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_5886_LIM__fun__gt__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [R3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X3))),R3)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,F2,X3))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_5887_LIM__fun__not__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X3))),R3)) )
               => ( aa(real,real,F2,X3) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_5888_LIM__fun__less__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [R3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X3))),R3)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,X3)),zero_zero(real))) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_5889_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))))
           => ( ? [D3: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
                  & ! [X2: A] :
                      ( ( ( X2 != A3 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A3))),D3)) )
                     => ( aa(A,B,F2,X2) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_sf(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_5890_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_ue(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_5891_isCont__arctan,axiom,
    ! [X: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arctan) ).

% isCont_arctan
tff(fact_5892_isCont__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(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_uf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_diff
tff(fact_5893_isCont__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_ug(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_minus
tff(fact_5894_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A3: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),aTP_Lamp_uh(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_5895_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),aTP_Lamp_ui(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_5896_isCont__cos_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_sy(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_cos'
tff(fact_5897_isCont__sin_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_sz(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_sin'
tff(fact_5898_isCont__exp_H,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: C,F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A3,top_top(set(C))),F2)
         => topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A3,top_top(set(C))),aTP_Lamp_sx(fun(C,A),fun(C,A),F2)) ) ) ).

% isCont_exp'
tff(fact_5899_isCont__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Z: A,N: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z,top_top(set(A))),aTP_Lamp_uj(nat,fun(A,A),N)) ) ).

% isCont_pochhammer
tff(fact_5900_continuous__at__within__powr,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [A3: C,S: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A3,S),F2)
         => ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A3,S),G)
           => ( ( aa(C,real,F2,A3) != zero_zero(real) )
             => topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A3,S),aa(fun(C,real),fun(C,real),aTP_Lamp_uk(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_5901_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S: set(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S),aTP_Lamp_ul(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_within_ln
tff(fact_5902_isCont__arsinh,axiom,
    ! [X: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arsinh(real)) ).

% isCont_arsinh
tff(fact_5903_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_um(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_5904_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_um(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_5905_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_un(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_5906_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K6: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
         => ( ! [H2: A] :
                ( ( H2 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H2)),K))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,H2))),aa(real,real,aa(real,fun(real,real),times_times(real),K6),real_V7770717601297561774m_norm(A,H2)))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_5907_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D6),topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_um(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_5908_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),ln_ln(real)) ) ).

% isCont_ln
tff(fact_5909_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_ue(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_5910_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_ui(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_5911_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_uo(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_5912_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),F2)
         => ( ( aa(A,A,cos(A),aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_sl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_5913_isCont__powr,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [A3: C,F2: fun(C,real),G: fun(C,real)] :
          ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A3,top_top(set(C))),F2)
         => ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A3,top_top(set(C))),G)
           => ( ( aa(C,real,F2,A3) != zero_zero(real) )
             => topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A3,top_top(set(C))),aa(fun(C,real),fun(C,real),aTP_Lamp_uk(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% isCont_powr
tff(fact_5914_isCont__ln_H,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,top_top(set(A))),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_ul(fun(A,real),fun(A,real),F2)) ) ) ) ).

% isCont_ln'
tff(fact_5915_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),F2)
         => ( ( aa(A,A,sin(A),aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_su(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_5916_continuous__at__within__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: C,A4: set(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A4),F2)
         => ( ( aa(A,A,cosh(A),aa(C,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A4),aTP_Lamp_up(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_5917_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A,X: A] :
          ( has_field_derivative(A,F2,L,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ? [G3: fun(A,A)] :
              ( ! [Z5: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,F2,Z5)),aa(A,A,F2,X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G3,Z5)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z5),X))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),G3)
              & ( aa(A,A,G3,X) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_5918_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( ! [X2: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X2))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
           => ? [M9: A] :
                ( ! [X3: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F2,X3)),M9)) )
                & ! [N8: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N8),M9))
                   => ? [X2: real] :
                        ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X2))
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N8),aa(real,A,F2,X2))) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_5919_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,cos(A),X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_5920_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_5921_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_5922_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,sin(A),X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_5923_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( aa(A,A,cosh(A),X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_5924_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)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S))
         => ( ! [X2: A] :
                ( ( X2 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X2)),S))
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),A3),X2),aa(A,A,F2,X2)) ) )
           => 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_5925_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A3: fun(nat,A),F2: fun(A,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S))
         => ( ! [X2: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X2)),S))
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),A3),X2),aa(A,A,F2,X2)) )
           => 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_5926_bij__int__encode,axiom,
    bij_betw(int,nat,nat_int_encode,top_top(set(int)),top_top(set(nat))) ).

% bij_int_encode
tff(fact_5927_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F2: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
         => ( summable(real,F2)
           => ( ! [H2: A,N2: nat] :
                  ( ( H2 != zero_zero(A) )
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H2)),K))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H2),N2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N2)),real_V7770717601297561774m_norm(A,H2)))) ) )
             => filterlim(A,B,aTP_Lamp_uq(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_5928_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( ( aa(A,A,cos(A),aa(A,A,F2,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_sl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_5929_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_5930_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)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,A3)))
             => ( ( aa(A,real,F2,A3) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),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_ur(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_5931_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_us(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_5932_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( ( aa(A,A,sin(A),aa(A,A,F2,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_su(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_5933_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A,C2: fun(nat,A),N: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_ut(fun(nat,A),fun(nat,fun(A,A)),C2),N)) ) ).

% isCont_polynom
tff(fact_5934_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_5935_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_5936_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_pu(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_5937_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)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,A3)))
             => ( ( aa(A,real,F2,A3) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),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_ur(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_5938_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_5939_isCont__inverse__function,axiom,
    ! [D2: real,X: real,G: fun(real,real),F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
     => ( ! [Z3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z3),X))),D2))
           => ( aa(real,real,G,aa(real,real,F2,Z3)) = Z3 ) )
       => ( ! [Z3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z3),X))),D2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_5940_GMVT_H,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),G: fun(real,real),G2: fun(real,real),F6: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( ! [Z3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),Z3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z3),B2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F2) ) )
       => ( ! [Z3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),Z3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z3),B2))
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),G) ) )
         => ( ! [Z3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Z3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),B2))
                 => has_field_derivative(real,G,aa(real,real,G2,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
           => ( ! [Z3: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Z3))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),B2))
                   => has_field_derivative(real,F2,aa(real,real,F6,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
             => ? [C3: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),C3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),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,G2,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_5941_isCont__powser_H,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [A3: A,F2: fun(A,Aa),C2: fun(nat,Aa),K6: Aa] :
          ( topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( summable(Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_uu(fun(nat,Aa),fun(Aa,fun(nat,Aa)),C2),K6))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(Aa,aa(A,Aa,F2,A3))),real_V7770717601297561774m_norm(Aa,K6)))
             => topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_uw(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_5942_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)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(nat,real,A3,zero_zero(nat))))
         => ! [N4: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A3))),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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))),N4))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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))),N4)),one_one(nat))))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_5943_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)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,A3,zero_zero(nat))),zero_zero(real)))
         => ! [N4: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A3))),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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))),N4)),one_one(nat)))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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))),N4)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_5944_summable__Leibniz_H_I4_J,axiom,
    ! [A3: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),suminf(real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A3))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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))),N)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_5945_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_uy(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_5946_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_uz(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_5947_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_va(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_5948_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ? [U2: fun(nat,A)] :
              ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,U2,N4)),X))
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_5949_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ? [U2: fun(nat,A)] :
              ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,U2,N4)))
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_5950_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_vb(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_5951_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_vb(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_5952_seq__offset__neg,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vc(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_5953_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_5954_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C2))
     => filterlim(nat,nat,aTP_Lamp_vd(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_5955_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C2))
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_5956_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_ve(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_5957_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A] : filterlim(nat,A,aTP_Lamp_vf(A,fun(nat,A),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_5958_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vg(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_5959_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X6: fun(nat,A),X: A,L: nat] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),L))
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vh(fun(nat,A),fun(nat,fun(nat,A)),X6),L),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_5960_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_vi(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_5961_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_vj(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_5962_nested__sequence__unique,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,N2)),aa(nat,real,F2,aa(nat,nat,suc,N2))))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,G,aa(nat,nat,suc,N2))),aa(nat,real,G,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,N2)),aa(nat,real,G,N2)))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_vk(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L2: real] :
                ( ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,N4)),L2))
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L2),at_top(nat))
                & ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L2),aa(nat,real,G,N4)))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_5963_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_vl(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_5964_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
     => filterlim(nat,real,aTP_Lamp_vm(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_5965_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_vn(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_5966_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_vo(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_5967_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,N2)),aa(nat,real,F2,aa(nat,nat,suc,N2))))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,N2)),L))
       => ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,N4)),E))) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_5968_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vp(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_5969_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vq(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_5970_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vr(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_5971_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_5972_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_vj(fun(nat,A),fun(nat,A),F2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_5973_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_vi(fun(nat,A),fun(nat,A),F2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_5974_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_vs(real,fun(real,fun(nat,real)),X),A3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_5975_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real)))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),C2)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_5976_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real)))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_5977_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => filterlim(nat,real,aTP_Lamp_vt(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_5978_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,F2,S)
        <=> filterlim(nat,A,aTP_Lamp_vu(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def'
tff(fact_5979_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),X: real] :
          ( filterlim(nat,real,aTP_Lamp_vv(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,X),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_5980_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_vw(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_5981_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L4: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N5)),L4))),R5)) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_5982_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L4: A] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No2),N2))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N2)),L4))),R3)) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L4),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_5983_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L4: A,R: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
           => ? [No3: nat] :
              ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N4))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N4)),L4))),R)) ) ) ) ) ).

% LIMSEQ_D
tff(fact_5984_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_5985_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim(nat,real,aTP_Lamp_vx(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_5986_tendsto__at__iff__sequentially,axiom,
    ! [C: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,C),A3: C,X: A,S: set(A)] :
          ( filterlim(A,C,F2,topolo7230453075368039082e_nhds(C,A3),topolo174197925503356063within(A,X,S))
        <=> ! [X7: fun(nat,A)] :
              ( ! [I2: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,X7,I2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))))
             => ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,X),at_top(nat))
               => filterlim(nat,C,aa(fun(nat,A),fun(nat,C),comp(A,C,nat,F2),X7),topolo7230453075368039082e_nhds(C,A3),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_5987_tendsto__power__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(B,nat),F4: filter(B),X: A] :
          ( filterlim(B,nat,F2,at_top(nat),F4)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
           => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_vy(fun(B,nat),fun(A,fun(B,A)),F2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_5988_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_vz(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_5989_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2)))))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_5990_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_ux(fun(nat,real),fun(nat,real),A3)) ) ) ).

% summable_Leibniz(1)
tff(fact_5991_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))
           => ( ! [N2: nat] : aa(nat,A,S,N2) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_wa(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_5992_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,aTP_Lamp_wb(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_5993_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X)))
         => filterlim(nat,A,aTP_Lamp_wc(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_5994_summable,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2)))
         => summable(real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A3)) ) ) ) ).

% summable
tff(fact_5995_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_wd(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K2: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_we(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_5996_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_wf(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K2: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_we(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_5997_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_wg(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_5998_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => filterlim(nat,real,aTP_Lamp_an(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_5999_summable__Leibniz_H_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2)))
         => filterlim(nat,real,aTP_Lamp_wg(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_6000_summable__Leibniz_H_I2_J,axiom,
    ! [A3: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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))),N)))),suminf(real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A3)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_6001_sums__alternating__upper__lower,axiom,
    ! [A3: fun(nat,real)] :
      ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2)))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2)))
       => ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L2: real] :
              ( ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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))),N4)))),L2))
              & filterlim(nat,real,aTP_Lamp_wg(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L2),at_top(nat))
              & ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L2),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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))),N4)),one_one(nat))))))
              & filterlim(nat,real,aTP_Lamp_wh(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_6002_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_wh(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_6003_summable__Leibniz_H_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2)))
         => filterlim(nat,real,aTP_Lamp_wh(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_6004_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_wi(fun(nat,A),fun(nat,A),F2),F4,at_top(nat))
    <=> filterlim(nat,A,F2,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_6005_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_wj(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_bounded_partials
tff(fact_6006_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,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_wk(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_6007_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,aTP_Lamp_wl(fun(nat,bool),fun(nat,bool),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_6008_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,bool),aTP_Lamp_wm(fun(A,B),fun(B,fun(A,bool)),F2),Z6),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_6009_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N7: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N7),N5))
             => pp(aa(A,bool,P,N5)) ) ) ) ).

% eventually_at_top_dense
tff(fact_6010_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),ord_less(A),C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_6011_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_qv(A,B)) ) ).

% bounded_linear_zero
tff(fact_6012_bounded__linear__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => real_V3181309239436604168linear(A,B,aTP_Lamp_qw(fun(A,B),fun(A,B),F2)) ) ) ).

% bounded_linear_minus
tff(fact_6013_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y2: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_wn(A,fun(A,A),Y2)) ) ).

% bounded_linear_divide
tff(fact_6014_bounded__linear__sub,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_6015_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),top_top(A)))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B3),top_top(A)))
                & ! [Z5: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B3),Z5))
                   => pp(aa(A,bool,P,Z5)) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_6016_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,bool),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_lessThan(A,X)))
        <=> ? [B3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B3),X))
              & ! [Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B3),Y4))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
                   => pp(aa(A,bool,P,Y4)) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_6017_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y2: A,X: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_lessThan(A,X)))
          <=> ? [B3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B3),X))
                & ! [Y4: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B3),Y4))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
                     => pp(aa(A,bool,P,Y4)) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_6018_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] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),C2),Z6))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_wo(fun(A,B),fun(B,fun(A,bool)),F2),Z6),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_6019_order__tendsto__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),X: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F4)
        <=> ( ! [L3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L3),X))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_wp(fun(B,A),fun(A,fun(B,bool)),F2),L3),F4) )
            & ! [U3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),U3))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_wq(fun(B,A),fun(A,fun(B,bool)),F2),U3),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_6020_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y2: A,F2: fun(B,A),F4: filter(B)] :
          ( ! [A6: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A6),Y2))
             => eventually(B,aa(A,fun(B,bool),aTP_Lamp_wp(fun(B,A),fun(A,fun(B,bool)),F2),A6),F4) )
         => ( ! [A6: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),A6))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_wq(fun(B,A),fun(A,fun(B,bool)),F2),A6),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y2),F4) ) ) ) ).

% order_tendstoI
tff(fact_6021_order__tendstoD_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),Y2: A,F4: filter(B),A3: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y2),F4)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),Y2))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_wp(fun(B,A),fun(A,fun(B,bool)),F2),A3),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_6022_order__tendstoD_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),Y2: A,F4: filter(B),A3: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y2),F4)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),A3))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_wq(fun(B,A),fun(A,fun(B,bool)),F2),A3),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_6023_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wr(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_6024_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B2: A,P: fun(A,bool)] :
          ( ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),set_or5935395276787703475ssThan(A,A3,B2)))
             => pp(aa(A,bool,P,X2)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => eventually(A,P,topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_6025_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,bool),A3: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,bool),aTP_Lamp_ws(fun(A,bool),fun(A,fun(A,bool)),P),A3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_6026_decreasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [L: A,F2: fun(B,A),F4: filter(B)] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_wt(A,fun(fun(B,A),fun(B,bool)),L),F2),F4)
         => ( ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),X2))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_wq(fun(B,A),fun(A,fun(B,bool)),F2),X2),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_6027_increasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),L: A,F4: filter(B)] :
          ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_wu(fun(B,A),fun(A,fun(B,bool)),F2),L),F4)
         => ( ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),L))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_wp(fun(B,A),fun(A,fun(B,bool)),F2),X2),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_6028_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( eventually(A,aTP_Lamp_wv(fun(A,real),fun(A,bool),F2),F4)
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_ww(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_6029_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L4: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),F4)
         => ( eventually(A,aa(B,fun(A,bool),aTP_Lamp_wx(fun(A,B),fun(B,fun(A,bool)),F2),L4),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L4,set_ord_lessThan(B,L4)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_6030_tendsto__arcosh__strong,axiom,
    ! [B: $tType,F2: fun(B,real),A3: real,F4: filter(B)] :
      ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,A3),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),A3))
       => ( eventually(B,aTP_Lamp_wy(fun(B,real),fun(B,bool),F2),F4)
         => filterlim(B,real,aTP_Lamp_tt(fun(B,real),fun(B,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A3)),F4) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_6031_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A),A3: A] :
          ( ! [X2: A,Y3: A] :
              ( pp(aa(A,bool,Q,X2))
             => ( pp(aa(A,bool,Q,Y3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y3))) ) ) )
         => ( ! [X2: B] :
                ( pp(aa(B,bool,P,X2))
               => ( aa(A,B,F2,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( pp(aa(B,bool,P,X2))
                 => pp(aa(A,bool,Q,aa(B,A,G,X2))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3)))
               => ( ! [B4: A] :
                      ( pp(aa(A,bool,Q,B4))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),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_6032_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,C),K6: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_wz(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),F2),G),K6),F4)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_6033_filterlim__at__withinI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),C2: A,F4: filter(B),A4: set(A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( eventually(B,aa(set(A),fun(B,bool),aa(A,fun(set(A),fun(B,bool)),aTP_Lamp_xa(fun(B,A),fun(A,fun(set(A),fun(B,bool))),F2),C2),A4),F4)
           => filterlim(B,A,F2,topolo174197925503356063within(A,C2,A4),F4) ) ) ) ).

% filterlim_at_withinI
tff(fact_6034_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) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
              & eventually(A,aTP_Lamp_xb(fun(A,real),fun(A,bool),F2),F4) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,aa(real,fun(real,real),powr(real),A3),B2)),F4) ) ) ) ).

% tendsto_powr'
tff(fact_6035_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_xb(fun(A,real),fun(A,bool),F2),F4)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,aa(real,fun(real,real),powr(real),A3),B2)),F4) ) ) ) ) ).

% tendsto_powr2
tff(fact_6036_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_xb(fun(A,real),fun(A,bool),F2),F4)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_6037_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)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_xc(fun(A,B),fun(B,fun(A,bool)),F2),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_6038_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)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_xd(fun(A,B),fun(B,fun(A,bool)),F2),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_6039_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_6040_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_xe(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F6),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_6041_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,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_xf(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_at_within
tff(fact_6042_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F6: fun(A,B),X: A,F2: fun(A,B),S: set(A)] :
          ( real_V3181309239436604168linear(A,B,F6)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_xg(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F6),X),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S))
           => has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivativeI
tff(fact_6043_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & ? [E3: fun(A,B)] :
                ( ! [H4: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F6,H4))),aa(A,B,E3,H4))
                & filterlim(A,real,aTP_Lamp_xh(fun(A,B),fun(A,real),E3),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% has_derivative_iff_Ex
tff(fact_6044_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,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_wk(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_within
tff(fact_6045_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D6: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,D6,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D6)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_xi(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D6),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_6046_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,N: nat,B5: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
             => eventually(A,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_xj(fun(nat,A),fun(nat,fun(real,fun(A,bool))),C2),N),B5),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_6047_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F6,F4)
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_xl(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F6),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_6048_cosh__real__at__top,axiom,
    filterlim(real,real,cosh(real),at_top(real),at_top(real)) ).

% cosh_real_at_top
tff(fact_6049_sinh__real__at__top,axiom,
    filterlim(real,real,sinh(real),at_top(real),at_top(real)) ).

% sinh_real_at_top
tff(fact_6050_arcosh__real__at__top,axiom,
    filterlim(real,real,arcosh(real),at_top(real),at_top(real)) ).

% arcosh_real_at_top
tff(fact_6051_arsinh__real__at__top,axiom,
    filterlim(real,real,arsinh(real),at_top(real),at_top(real)) ).

% arsinh_real_at_top
tff(fact_6052_lhopital__at__top__at__top,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F6: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,A3,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,A3,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G2),at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A3,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_6053_lhopital__left__at__top__at__top,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F6: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G2),at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_6054_ln__at__top,axiom,
    filterlim(real,real,ln_ln(real),at_top(real),at_top(real)) ).

% ln_at_top
tff(fact_6055_exp__at__top,axiom,
    filterlim(real,real,exp(real),at_top(real),at_top(real)) ).

% exp_at_top
tff(fact_6056_filterlim__int__sequentially,axiom,
    filterlim(nat,int,semiring_1_of_nat(int),at_top(int),at_top(nat)) ).

% filterlim_int_sequentially
tff(fact_6057_lhospital__at__top__at__top,axiom,
    ! [G: fun(real,real),G2: fun(real,real),F2: fun(real,real),F6: fun(real,real),X: real] :
      ( filterlim(real,real,G,at_top(real),at_top(real))
     => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G2),at_top(real))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),at_top(real))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),at_top(real))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F6),topolo7230453075368039082e_nhds(real,X),at_top(real))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,X),at_top(real)) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_6058_lhopital__at__top,axiom,
    ! [G: fun(real,real),X: real,G2: fun(real,real),F2: fun(real,real),F6: fun(real,real),Y2: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G2),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F6),topolo7230453075368039082e_nhds(real,Y2),topolo174197925503356063within(real,X,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y2),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_6059_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),X: real,G2: fun(real,real),F2: fun(real,real),F6: fun(real,real),Y2: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
     => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F6),topolo7230453075368039082e_nhds(real,Y2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_6060_tendsto__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => filterlim(nat,A,semiring_1_of_nat(A),at_infinity(A),at_top(nat)) ) ).

% tendsto_of_nat
tff(fact_6061_filterlim__pow__at__top,axiom,
    ! [A: $tType,N: nat,F2: fun(A,real),F4: filter(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_se(nat,fun(fun(A,real),fun(A,real)),N),F2),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_6062_tanh__real__at__top,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(real)) ).

% tanh_real_at_top
tff(fact_6063_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_xq(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_ue(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_6064_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( filterlim(A,real,G,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% real_tendsto_divide_at_top
tff(fact_6065_artanh__real__at__left__1,axiom,
    filterlim(real,real,artanh(real),at_top(real),topolo174197925503356063within(real,one_one(real),set_ord_lessThan(real,one_one(real)))) ).

% artanh_real_at_left_1
tff(fact_6066_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_uh(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_6067_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_ui(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_6068_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,F2: fun(int,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_xs(fun(int,A),fun(nat,A),F2),F4,at_top(nat))
     => filterlim(int,A,F2,F4,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_6069_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,real,F4,G)
           => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_xt(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_powr
tff(fact_6070_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_ul(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_ln
tff(fact_6071_tendsto__inverse__0,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_infinity(A)) ) ).

% tendsto_inverse_0
tff(fact_6072_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,F2: fun(A,real),F4: filter(A)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),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_xu(real,fun(fun(A,real),fun(A,real)),S),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_neg_powr
tff(fact_6073_tendsto__mult__filterlim__at__infinity,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),C2: A,F4: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( ( C2 != zero_zero(A) )
           => ( filterlim(B,A,G,at_infinity(A),F4)
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xv(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),at_infinity(A),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_6074_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_xw(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_6075_tendsto__divide__0,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(C,A),C2: A,F4: filter(C),G: fun(C,A)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( filterlim(C,A,G,at_infinity(A),F4)
           => filterlim(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_xx(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_6076_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),N: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_xy(fun(A,B),fun(nat,fun(A,B)),F2),N),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_6077_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( aa(A,A,cos(A),aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xz(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_sl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_6078_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( aa(A,A,sin(A),aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xz(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_su(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_6079_continuous__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F4,F2)
         => ( ( aa(A,A,cosh(A),aa(C,A,F2,topolo3827282254853284352ce_Lim(C,C,F4,aTP_Lamp_ya(C,C)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,F4,aTP_Lamp_up(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_tanh
tff(fact_6080_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A)))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_ww(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_6081_filterlim__inverse__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),at_infinity(A),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% filterlim_inverse_at_infinity
tff(fact_6082_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)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => ( eventually(A,aTP_Lamp_yb(fun(A,real),fun(A,bool),G),F4)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_6083_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_yc(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_6084_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_yd(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F4)
        <=> filterlim(A,B,G,at_infinity(B),F4) ) ) ).

% filterlim_inverse_at_iff
tff(fact_6085_lhopital,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G2: fun(real,real),F6: fun(real,real),F4: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G2),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,X,top_top(set(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,X,top_top(set(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F6),F4,topolo174197925503356063within(real,X,top_top(set(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F4,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_6086_lhopital__left,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G2: fun(real,real),F6: fun(real,real),F4: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
       => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
         => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F6),F4,topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F4,topolo174197925503356063within(real,X,set_ord_lessThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_6087_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)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A)))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A)))))
                 => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_ur(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_6088_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_ye(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_6089_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_po(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_6090_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A)))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_yf(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_artanh
tff(fact_6091_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_6092_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_6093_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X)))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),X),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_6094_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_6095_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_yg(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_6096_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [X: A,S3: set(A),F2: fun(A,B),F6: fun(A,B)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
         => ( topolo1002775350975398744n_open(A,S3)
           => ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S3))
            <=> ( real_V3181309239436604168linear(A,B,F6)
                & ? [E3: fun(A,B)] :
                    ( ! [H4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H4)),S3))
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F6,H4))),aa(A,B,E3,H4)) ) )
                    & filterlim(A,real,aTP_Lamp_xh(fun(A,B),fun(A,real),E3),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ) ).

% has_derivative_at_within_iff_Ex
tff(fact_6097_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E2: real,F6: fun(A,B),S: set(A),X: A,F2: fun(A,B),H5: fun(A,real)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
         => ( real_V3181309239436604168linear(A,B,F6)
           => ( ! [Y3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),S))
                 => ( ( Y3 != X )
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y3,X)),E2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),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,Y3)),aa(A,B,F2,X))),aa(A,B,F6,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),X))))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),X)))),aa(A,real,H5,Y3))) ) ) )
             => ( filterlim(A,real,H5,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S))
               => has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_6098_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V557655796197034286t_dist(A,zero_zero(A),X) = real_V7770717601297561774m_norm(A,X) ) ).

% dist_0_norm
tff(fact_6099_dist__diff_I1_J,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A] : real_V557655796197034286t_dist(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = real_V7770717601297561774m_norm(A,B2) ) ).

% dist_diff(1)
tff(fact_6100_dist__diff_I2_J,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2),A3) = real_V7770717601297561774m_norm(A,B2) ) ).

% dist_diff(2)
tff(fact_6101_dist__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: real,A3: A,Y2: real] : real_V557655796197034286t_dist(A,real_V8093663219630862766scaleR(A,X,A3),real_V8093663219630862766scaleR(A,Y2,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y2))),real_V7770717601297561774m_norm(A,A3)) ) ).

% dist_scaleR
tff(fact_6102_dist__norm,axiom,
    ! [A: $tType] :
      ( real_V6936659425649961206t_norm(A)
     => ! [X: A,Y2: A] : real_V557655796197034286t_dist(A,X,Y2) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) ) ).

% dist_norm
tff(fact_6103_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V7770717601297561774m_norm(A,X) = real_V557655796197034286t_dist(A,X,zero_zero(A)) ) ).

% norm_conv_dist
tff(fact_6104_dist__real__def,axiom,
    ! [X: real,Y2: real] : real_V557655796197034286t_dist(real,X,Y2) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y2)) ).

% dist_real_def
tff(fact_6105_dist__complex__def,axiom,
    ! [X: complex,Y2: complex] : real_V557655796197034286t_dist(complex,X,Y2) = real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y2)) ).

% dist_complex_def
tff(fact_6106_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),X: A,Y2: A] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
             => ? [B4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B4))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,X,B4)),S3)) ) ) ) ) ) ).

% open_right
tff(fact_6107_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),X: A,Y2: A] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
             => ? [B4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),X))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B4,X)),S3)) ) ) ) ) ) ).

% open_left
tff(fact_6108_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [A3: A,B2: A,C2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V557655796197034286t_dist(A,A3,B2)),real_V557655796197034286t_dist(A,B2,C2)))),real_V557655796197034286t_dist(A,A3,C2))) ) ).

% abs_dist_diff_le
tff(fact_6109_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_yh(product_prod(A,A),bool))) ) ).

% open_superdiagonal
tff(fact_6110_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_yi(product_prod(A,A),bool))) ) ).

% open_subdiagonal
tff(fact_6111_dist__of__int,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [M2: int,N: int] : real_V557655796197034286t_dist(A,aa(int,A,ring_1_of_int(A),M2),aa(int,A,ring_1_of_int(A),N)) = aa(int,real,ring_1_of_int(real),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),M2),N))) ) ).

% dist_of_int
tff(fact_6112_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y2: A,E2: real,X23: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,Y2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X23,Y2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X23)),E2)) ) ) ) ).

% dist_triangle_half_l
tff(fact_6113_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y2: A,X15: A,E2: real,X23: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y2,X15)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y2,X23)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X23)),E2)) ) ) ) ).

% dist_triangle_half_r
tff(fact_6114_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,X23: A,E2: real,X32: A,X42: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X23)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X23,X32)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X42)),E2)) ) ) ) ) ).

% dist_triangle_third
tff(fact_6115_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A,S3: set(A),T5: set(A),U4: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
         => ( topolo1002775350975398744n_open(A,S3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T5),S3)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U4),S3)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,X,T5) = topolo174197925503356063within(A,X,U4) ) ) ) ) ) ).

% at_within_nhd
tff(fact_6116_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M8: nat] :
                ! [M3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M3))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M3),aa(nat,A,X6,N2))),E)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI'
tff(fact_6117_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [M7: nat] :
                ! [M5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),M5))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F2,M5),aa(nat,A,F2,N5))),E3)) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_6118_dist__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [M2: nat,N: nat] : real_V557655796197034286t_dist(A,aa(nat,A,semiring_1_of_nat(A),M2),aa(nat,A,semiring_1_of_nat(A),N)) = aa(int,real,ring_1_of_int(real),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N)))) ) ).

% dist_of_nat
tff(fact_6119_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [J3: nat] :
            ? [M7: nat] :
            ! [M5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),M5))
             => ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M5),aa(nat,A,X6,N5))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_6120_tendsto__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A3: A,S3: set(A),F2: fun(A,D),L4: D] :
          ( nO_MATCH(C,A,zero_zero(C),A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),S3))
           => ( topolo1002775350975398744n_open(A,S3)
             => ( filterlim(A,D,F2,topolo7230453075368039082e_nhds(D,L4),topolo174197925503356063within(A,A3,S3))
              <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_ub(A,fun(fun(A,D),fun(A,D)),A3),F2),topolo7230453075368039082e_nhds(D,L4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_6121_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L4: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),No))
                  & ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N5),L4)),R5)) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_6122_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F2: fun(real,real),F4: filter(real)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yj(nat,fun(fun(real,real),fun(real,real)),N),F2),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_6123_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_yk(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_6124_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_ord_greaterThan(A,K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),I)) ) ) ).

% greaterThan_iff
tff(fact_6125_Compl__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),set_ord_greaterThan(A,K)) = set_ord_atMost(A,K) ) ).

% Compl_greaterThan
tff(fact_6126_Compl__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),set_ord_atMost(A,K)) = set_ord_greaterThan(A,K) ) ).

% Compl_atMost
tff(fact_6127_image__uminus__greaterThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : image(A,A,uminus_uminus(A),set_ord_greaterThan(A,X)) = set_ord_lessThan(A,aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_greaterThan
tff(fact_6128_image__uminus__lessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : image(A,A,uminus_uminus(A),set_ord_lessThan(A,X)) = set_ord_greaterThan(A,aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_lessThan
tff(fact_6129_ln__at__0,axiom,
    filterlim(real,real,ln_ln(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% ln_at_0
tff(fact_6130_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_greaterThan(A,L) = collect(A,aa(A,fun(A,bool),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_6131_artanh__real__at__right__1,axiom,
    filterlim(real,real,artanh(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),one_one(real)),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),one_one(real))))) ).

% artanh_real_at_right_1
tff(fact_6132_sinh__real__at__bot,axiom,
    filterlim(real,real,sinh(real),at_bot(real),at_bot(real)) ).

% sinh_real_at_bot
tff(fact_6133_arsinh__real__at__bot,axiom,
    filterlim(real,real,arsinh(real),at_bot(real),at_bot(real)) ).

% arsinh_real_at_bot
tff(fact_6134_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A),A3: A] :
          ( ! [X2: A,Y3: A] :
              ( pp(aa(A,bool,Q,X2))
             => ( pp(aa(A,bool,Q,Y3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y3))) ) ) )
         => ( ! [X2: B] :
                ( pp(aa(B,bool,P,X2))
               => ( aa(A,B,F2,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( pp(aa(B,bool,P,X2))
                 => pp(aa(A,bool,Q,aa(B,A,G,X2))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3)))
               => ( ! [B4: A] :
                      ( pp(aa(A,bool,Q,B4))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B4)) )
                 => ( 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_6135_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,Y2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_greaterThan(A,X)))
          <=> ? [B3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B3))
                & ! [Y4: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),B3))
                     => pp(aa(A,bool,P,Y4)) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_6136_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,bool),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_greaterThan(A,X)))
        <=> ? [B3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B3))
              & ! [Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),B3))
                   => pp(aa(A,bool,P,Y4)) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_6137_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6138_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N7: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N5),N7))
             => pp(aa(A,bool,P,N5)) ) ) ) ).

% eventually_at_bot_dense
tff(fact_6139_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : eventually(A,aa(A,fun(A,bool),ord_less(A),X),topolo174197925503356063within(A,X,set_ord_greaterThan(A,X))) ) ).

% eventually_at_right_less
tff(fact_6140_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_yl(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_6141_cosh__real__at__bot,axiom,
    filterlim(real,real,cosh(real),at_top(real),at_bot(real)) ).

% cosh_real_at_bot
tff(fact_6142_lhopital__right__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F6: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G2),at_bot(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_6143_filterlim__uminus__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F4)
    <=> filterlim(A,real,aTP_Lamp_ym(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ).

% filterlim_uminus_at_top
tff(fact_6144_filterlim__uminus__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_bot(real),F4)
    <=> filterlim(A,real,aTP_Lamp_ym(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ).

% filterlim_uminus_at_bot
tff(fact_6145_filterlim__at__top__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_yn(fun(real,A),fun(real,A),F2),F4,at_bot(real)) ) ).

% filterlim_at_top_mirror
tff(fact_6146_filterlim__at__bot__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_bot(real))
    <=> filterlim(real,A,aTP_Lamp_yn(fun(real,A),fun(real,A),F2),F4,at_top(real)) ) ).

% filterlim_at_bot_mirror
tff(fact_6147_filterlim__uminus__at__top__at__bot,axiom,
    filterlim(real,real,uminus_uminus(real),at_top(real),at_bot(real)) ).

% filterlim_uminus_at_top_at_bot
tff(fact_6148_filterlim__uminus__at__bot__at__top,axiom,
    filterlim(real,real,uminus_uminus(real),at_bot(real),at_top(real)) ).

% filterlim_uminus_at_bot_at_top
tff(fact_6149_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ? [A6: A,B4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),set_ord_lessThan(A,A6)))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),set_ord_greaterThan(A,B4)))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,A6)),set_ord_greaterThan(A,B4)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_6150_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B2: A,P: fun(A,bool)] :
          ( ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),set_or5935395276787703475ssThan(A,A3,B2)))
             => pp(aa(A,bool,P,X2)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => eventually(A,P,topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% eventually_at_rightI
tff(fact_6151_filterlim__at__left__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A),A3: real] :
      ( filterlim(real,A,F2,F4,topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
    <=> filterlim(real,A,aTP_Lamp_yn(fun(real,A),fun(real,A),F2),F4,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A3),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A3)))) ) ).

% filterlim_at_left_to_right
tff(fact_6152_eventually__at__left__to__right,axiom,
    ! [P: fun(real,bool),A3: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
    <=> eventually(real,aTP_Lamp_yo(fun(real,bool),fun(real,bool),P),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A3),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A3)))) ) ).

% eventually_at_left_to_right
tff(fact_6153_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_6154_exp__at__bot,axiom,
    filterlim(real,real,exp(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_bot(real)) ).

% exp_at_bot
tff(fact_6155_filterlim__times__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(B,A),P2: A,F12: filter(B),C2: A,L: A] :
          ( filterlim(B,A,F2,topolo174197925503356063within(A,P2,set_ord_greaterThan(A,P2)),F12)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( ( L = aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2) )
             => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_yp(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo174197925503356063within(A,L,set_ord_greaterThan(A,L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_6156_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,bool),aTP_Lamp_yq(fun(A,B),fun(B,fun(A,bool)),F2),Z6),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_6157_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L4: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),F4)
         => ( eventually(A,aa(B,fun(A,bool),aTP_Lamp_yr(fun(A,B),fun(B,fun(A,bool)),F2),L4),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L4,set_ord_greaterThan(B,L4)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_6158_tanh__real__at__bot,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),one_one(real))),at_bot(real)) ).

% tanh_real_at_bot
tff(fact_6159_tendsto__arcosh__at__left__1,axiom,
    filterlim(real,real,arcosh(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,one_one(real),set_ord_greaterThan(real,one_one(real)))) ).

% tendsto_arcosh_at_left_1
tff(fact_6160_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] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z6),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_ys(fun(A,B),fun(B,fun(A,bool)),F2),Z6),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_6161_lhopital__right__at__top__at__top,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F6: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G2),at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_6162_lhopital__right__0,axiom,
    ! [F0: fun(real,real),G0: fun(real,real),G2: fun(real,real),F6: fun(real,real),F4: filter(real)] :
      ( filterlim(real,real,F0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( filterlim(real,real,G0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G0),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F0),F6),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G0),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F6),F4,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F0),G0),F4,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_6163_lhopital__right,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G2: fun(real,real),F6: fun(real,real),F4: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
       => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
         => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F6),F4,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F4,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_6164_lhopital__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F6: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A3,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,A3,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,A3,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G2),at_bot(real),topolo174197925503356063within(real,A3,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A3,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_6165_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F2: fun(real,real),F4: filter(real)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yj(nat,fun(fun(real,real),fun(real,real)),N),F2),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_6166_lhopital__right__at__top,axiom,
    ! [G: fun(real,real),X: real,G2: fun(real,real),F2: fun(real,real),F6: fun(real,real),Y2: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
     => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F6),topolo7230453075368039082e_nhds(real,Y2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_6167_lhopital__right__0__at__top,axiom,
    ! [G: fun(real,real),G2: fun(real,real),F2: fun(real,real),F6: fun(real,real),X: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( eventually(real,aTP_Lamp_xo(fun(real,real),fun(real,bool),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F6),topolo7230453075368039082e_nhds(real,X),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,X),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_6168_lhopital__left__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F6: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F6),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G2),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G2),at_bot(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_6169_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_6170_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( ! [F3: fun(nat,A)] :
                ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(nat,A,F3,N4)))
               => ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N4)),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,bool),aTP_Lamp_yt(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_6171_GMVT,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),G: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( ! [X2: real] :
            ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X2))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
       => ( ! [X2: real] :
              ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X2))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),B2)) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X2,top_top(set(real)))) )
         => ( ! [X2: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X2))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X2,top_top(set(real))),G) )
           => ( ! [X2: real] :
                  ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X2))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),B2)) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X2,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C3,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),C3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),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_6172_differentiable__cmult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [C2: A,Q5: fun(B,A),T2: B] :
          ( differentiable(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yu(A,fun(fun(B,A),fun(B,A)),C2),Q5),topolo174197925503356063within(B,T2,top_top(set(B))))
        <=> ( ( C2 = zero_zero(A) )
            | differentiable(B,A,Q5,topolo174197925503356063within(B,T2,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_6173_differentiable__cmult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [Q5: fun(B,A),C2: A,T2: B] :
          ( differentiable(B,A,aa(A,fun(B,A),aTP_Lamp_yv(fun(B,A),fun(A,fun(B,A)),Q5),C2),topolo174197925503356063within(B,T2,top_top(set(B))))
        <=> ( ( C2 = zero_zero(A) )
            | differentiable(B,A,Q5,topolo174197925503356063within(B,T2,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_6174_differentiable__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( differentiable(A,B,F2,F4)
         => differentiable(A,B,aTP_Lamp_qw(fun(A,B),fun(A,B),F2),F4) ) ) ).

% differentiable_minus
tff(fact_6175_differentiable__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F4)
         => ( differentiable(A,B,G,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% differentiable_diff
tff(fact_6176_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N5))),aa(nat,A,F2,N5))) ) ) ).

% decseq_Suc_iff
tff(fact_6177_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N2))),aa(nat,A,X6,N2)))
         => order_antimono(nat,A,X6) ) ) ).

% decseq_SucI
tff(fact_6178_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A4)
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A4,aa(nat,nat,suc,I))),aa(nat,A,A4,I))) ) ) ).

% decseq_SucD
tff(fact_6179_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% differentiable_divide
tff(fact_6180_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,B,F2,X) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_yx(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% differentiable_inverse
tff(fact_6181_max__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),X: A,Y2: A] :
          ( order_antimono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)) ) ) ) ).

% max_of_antimono
tff(fact_6182_min__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),X: A,Y2: A] :
          ( order_antimono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)) ) ) ) ).

% min_of_antimono
tff(fact_6183_greaterThan__0,axiom,
    set_ord_greaterThan(nat,zero_zero(nat)) = image(nat,nat,suc,top_top(set(nat))) ).

% greaterThan_0
tff(fact_6184_greaterThan__Suc,axiom,
    ! [K: nat] : set_ord_greaterThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_greaterThan(nat,K)),aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_6185_tendsto__at__right__sequentially,axiom,
    ! [C: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: B,B2: B,X6: fun(B,C),L4: C] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A3),B2))
         => ( ! [S7: fun(nat,B)] :
                ( ! [N4: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A3),aa(nat,B,S7,N4)))
               => ( ! [N4: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(nat,B,S7,N4)),B2))
                 => ( order_antimono(nat,B,S7)
                   => ( filterlim(nat,B,S7,topolo7230453075368039082e_nhds(B,A3),at_top(nat))
                     => filterlim(nat,C,aa(fun(nat,B),fun(nat,C),aTP_Lamp_yy(fun(B,C),fun(fun(nat,B),fun(nat,C)),X6),S7),topolo7230453075368039082e_nhds(C,L4),at_top(nat)) ) ) ) )
           => filterlim(B,C,X6,topolo7230453075368039082e_nhds(C,L4),topolo174197925503356063within(B,A3,set_ord_greaterThan(B,A3))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_6186_Bfun__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A3: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( A3 != zero_zero(A) )
           => bfun(B,A,aTP_Lamp_sp(fun(B,A),fun(B,A),F2),F4) ) ) ) ).

% Bfun_inverse
tff(fact_6187_MVT,axiom,
    ! [A3: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
       => ( ! [X2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),B2))
               => differentiable(real,real,F2,topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
         => ? [L2: real,Z3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Z3))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),B2))
              & has_field_derivative(real,F2,L2,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)),L2) ) ) ) ) ) ).

% MVT
tff(fact_6188_continuous__on__ln,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
               => ( aa(A,real,F2,X2) != zero_zero(real) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_yz(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_ln
tff(fact_6189_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
               => ( aa(A,B,F2,X2) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_za(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_6190_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)
           => ( ! [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
                 => ( aa(A,B,G,X2) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_zb(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_6191_continuous__on__op__minus,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [S: set(A),X: A] : topolo81223032696312382ous_on(A,A,S,aa(A,fun(A,A),minus_minus(A),X)) ) ).

% continuous_on_op_minus
tff(fact_6192_continuous__on__diff,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo1633459387980952147up_add(B) )
     => ! [S: set(D),F2: fun(D,B),G: fun(D,B)] :
          ( topolo81223032696312382ous_on(D,B,S,F2)
         => ( topolo81223032696312382ous_on(D,B,S,G)
           => topolo81223032696312382ous_on(D,B,S,aa(fun(D,B),fun(D,B),aTP_Lamp_zc(fun(D,B),fun(fun(D,B),fun(D,B)),F2),G)) ) ) ) ).

% continuous_on_diff
tff(fact_6193_continuous__on__arsinh_H,axiom,
    ! [A4: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A4,F2)
     => topolo81223032696312382ous_on(real,real,A4,aTP_Lamp_zd(fun(real,real),fun(real,real),F2)) ) ).

% continuous_on_arsinh'
tff(fact_6194_continuous__on__arctan,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_ze(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_arctan
tff(fact_6195_continuous__on__cosh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A4: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,A4,F2)
         => topolo81223032696312382ous_on(C,A,A4,aTP_Lamp_zf(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_on_cosh
tff(fact_6196_continuous__on__exp,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,S,F2)
         => topolo81223032696312382ous_on(C,A,S,aTP_Lamp_zg(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_on_exp
tff(fact_6197_continuous__on__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_sz(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_sin
tff(fact_6198_continuous__on__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_sy(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_cos
tff(fact_6199_continuous__on__sinh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A4: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,A4,F2)
         => topolo81223032696312382ous_on(C,A,A4,aTP_Lamp_zh(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_on_sinh
tff(fact_6200_continuous__on__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A4: set(A),N: nat] : topolo81223032696312382ous_on(A,A,A4,aTP_Lamp_uj(nat,fun(A,A),N)) ) ).

% continuous_on_pochhammer
tff(fact_6201_continuous__on__pochhammer_H,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & topolo4958980785337419405_space(C) )
     => ! [S: set(C),F2: fun(C,A),N: nat] :
          ( topolo81223032696312382ous_on(C,A,S,F2)
         => topolo81223032696312382ous_on(C,A,S,aa(nat,fun(C,A),aTP_Lamp_zi(fun(C,A),fun(nat,fun(C,A)),F2),N)) ) ) ).

% continuous_on_pochhammer'
tff(fact_6202_continuous__on__arsinh,axiom,
    ! [A4: set(real)] : topolo81223032696312382ous_on(real,real,A4,arsinh(real)) ).

% continuous_on_arsinh
tff(fact_6203_continuous__on__minus,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo1633459387980952147up_add(B) )
     => ! [S: set(C),F2: fun(C,B)] :
          ( topolo81223032696312382ous_on(C,B,S,F2)
         => topolo81223032696312382ous_on(C,B,S,aTP_Lamp_zj(fun(C,B),fun(C,B),F2)) ) ) ).

% continuous_on_minus
tff(fact_6204_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
               => ( aa(A,B,F2,X2) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_zk(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_6205_continuous__on__sin__real,axiom,
    ! [A3: real,B2: real] : topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),sin(real)) ).

% continuous_on_sin_real
tff(fact_6206_continuous__on__cos__real,axiom,
    ! [A3: real,B2: real] : topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),cos(real)) ).

% continuous_on_cos_real
tff(fact_6207_continuous__on__powr,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [S: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo81223032696312382ous_on(C,real,S,F2)
         => ( topolo81223032696312382ous_on(C,real,S,G)
           => ( ! [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),S))
                 => ( aa(C,real,F2,X2) != zero_zero(real) ) )
             => topolo81223032696312382ous_on(C,real,S,aa(fun(C,real),fun(C,real),aTP_Lamp_zl(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr
tff(fact_6208_Bseq__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),X6),at_top(nat))
        <=> bfun(nat,A,X6,at_top(nat)) ) ) ).

% Bseq_minus_iff
tff(fact_6209_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bq(fun(nat,A),fun(nat,A),F2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_6210_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_zm(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_6211_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
               => ( aa(A,A,cos(A),aa(A,A,F2,X2)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_sl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_6212_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
               => ( aa(A,A,sin(A),aa(A,A,F2,X2)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_su(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_6213_continuous__on__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A4: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,A4,F2)
         => ( ! [X2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),A4))
               => ( aa(A,A,cosh(A),aa(C,A,F2,X2)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(C,A,A4,aTP_Lamp_zn(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_6214_continuous__on__arcosh_H,axiom,
    ! [A4: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A4,F2)
     => ( ! [X2: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),A4))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,F2,X2))) )
       => topolo81223032696312382ous_on(real,real,A4,aTP_Lamp_zo(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_arcosh'
tff(fact_6215_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_bl(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_6216_continuous__on__powr_H,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [S: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo81223032696312382ous_on(C,real,S,F2)
         => ( topolo81223032696312382ous_on(C,real,S,G)
           => ( ! [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),S))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(C,real,F2,X2)))
                    & ( ( aa(C,real,F2,X2) = zero_zero(real) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(C,real,G,X2))) ) ) )
             => topolo81223032696312382ous_on(C,real,S,aa(fun(C,real),fun(C,real),aTP_Lamp_zl(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_6217_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)
           => ( ! [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,X2))) )
             => ( ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
                   => ( aa(A,real,F2,X2) != one_one(real) ) )
               => ( ! [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X2))) )
                 => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_zp(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_6218_continuous__on__arccos_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arccos) ).

% continuous_on_arccos'
tff(fact_6219_continuous__on__arcsin_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arcsin) ).

% continuous_on_arcsin'
tff(fact_6220_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X2)))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,F2,X2)),one_one(real))) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_zq(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arccos
tff(fact_6221_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X2)))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,F2,X2)),one_one(real))) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_zr(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arcsin
tff(fact_6222_continuous__on__artanh,axiom,
    ! [A4: set(real)] :
      ( pp(aa(set(real),bool,aa(set(real),fun(set(real),bool),ord_less_eq(set(real)),A4),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))))
     => topolo81223032696312382ous_on(real,real,A4,artanh(real)) ) ).

% continuous_on_artanh
tff(fact_6223_continuous__on__artanh_H,axiom,
    ! [A4: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A4,F2)
     => ( ! [X2: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),A4))
           => pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,F2,X2)),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))) )
       => topolo81223032696312382ous_on(real,real,A4,aTP_Lamp_zs(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_artanh'
tff(fact_6224_mvt,axiom,
    ! [A3: real,B2: real,F2: fun(real,real),F6: fun(real,fun(real,real))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F2)
       => ( ! [X2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X2),B2))
               => has_derivative(real,real,F2,aa(real,fun(real,real),F6,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
         => ~ ! [Xi: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Xi))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),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_6225_continuous__on__of__int__floor,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_zt(A,B)) ) ).

% continuous_on_of_int_floor
tff(fact_6226_continuous__on__of__int__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_zu(A,B)) ) ).

% continuous_on_of_int_ceiling
tff(fact_6227_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N7: nat] :
            ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% Bseq_iff1a
tff(fact_6228_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N7: nat] :
            ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% Bseq_iff
tff(fact_6229_Bseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => bfun(nat,real,aa(real,fun(nat,real),power_power(real),X),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_6230_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)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_6231_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)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_6232_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)))
           => ( ! [X2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X2))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),B2))
                   => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X2)),topolo174197925503356063within(A,X2,top_top(set(A)))) ) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B2),F2) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_6233_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K3))
              & ? [X4: A] :
                ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N5)),aa(A,A,uminus_uminus(A),X4)))),K3)) ) ) ) ).

% Bseq_iff2
tff(fact_6234_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K3))
              & ? [N7: nat] :
                ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N5)),aa(A,A,uminus_uminus(A),aa(nat,A,X6,N7))))),K3)) ) ) ) ).

% Bseq_iff3
tff(fact_6235_eventually__filtercomap__at__topological,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,bool),F2: fun(A,B),A4: B,B5: set(B)] :
          ( eventually(A,P,filtercomap(A,B,F2,topolo174197925503356063within(B,A4,B5)))
        <=> ? [S8: set(B)] :
              ( topolo1002775350975398744n_open(B,S8)
              & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S8))
              & ! [X4: A] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S8),B5)),aa(set(B),set(B),insert(B,A4),bot_bot(set(B))))))
                 => pp(aa(A,bool,P,X4)) ) ) ) ) ).

% eventually_filtercomap_at_topological
tff(fact_6236_sequentially__imp__eventually__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,A3: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( ! [F3: fun(nat,A)] :
                ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(nat,A,F3,N4)))
               => ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N4)),A3))
                 => ( order_mono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_yt(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_6237_decseq__eq__incseq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X6: fun(nat,A)] :
          ( order_antimono(nat,A,X6)
        <=> order_mono(nat,A,aTP_Lamp_he(fun(nat,A),fun(nat,A),X6)) ) ) ).

% decseq_eq_incseq
tff(fact_6238_min__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),M2: A,N: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,M2)),aa(A,B,F2,N)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)) ) ) ) ).

% min_of_mono
tff(fact_6239_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_mono(nat,A,aTP_Lamp_zv(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_6240_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),A4: A,B5: A,N: nat] :
          ( order_mono(A,A,F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),B5))
           => pp(aa(A,bool,aa(A,fun(A,bool),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)),N),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)),N),F2),B5))) ) ) ) ).

% funpow_mono
tff(fact_6241_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y2: A] :
          ( order_mono(A,B,F2)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) ) ) ) ).

% mono_invE
tff(fact_6242_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_mono(nat,A,F2)
        <=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N5)),aa(nat,A,F2,aa(nat,nat,suc,N5)))) ) ) ).

% incseq_Suc_iff
tff(fact_6243_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,X6,aa(nat,nat,suc,N2))))
         => order_mono(nat,A,X6) ) ) ).

% incseq_SucI
tff(fact_6244_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: fun(nat,A),I: nat] :
          ( order_mono(nat,A,A4)
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A4,I)),aa(nat,A,A4,aa(nat,nat,suc,I)))) ) ) ).

% incseq_SucD
tff(fact_6245_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A] : order_mono(A,A,aa(A,fun(A,A),plus_plus(A),A3)) ) ).

% mono_add
tff(fact_6246_mono__Suc,axiom,
    order_mono(nat,nat,suc) ).

% mono_Suc
tff(fact_6247_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y2: A] :
          ( order_mono(A,B,F2)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2)) ) ) ) ).

% mono_strict_invE
tff(fact_6248_max__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),M2: A,N: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,M2)),aa(A,B,F2,N)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),M2),N)) ) ) ) ).

% max_of_mono
tff(fact_6249_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),N: nat] :
          ( order_mono(A,A,F2)
         => order_mono(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)) ) ) ).

% mono_pow
tff(fact_6250_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F2: fun(A,B),A4: A,B5: A] :
          ( order_mono(A,B,F2)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B5))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A4)),aa(A,B,F2,B5)))) ) ) ).

% mono_inf
tff(fact_6251_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => order_mono(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N)) ) ).

% mono_times_nat
tff(fact_6252_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => order_mono(A,A,aa(A,fun(A,A),times_times(A),A3)) ) ) ).

% mono_mult
tff(fact_6253_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F2: fun(A,B),M2: A,N: A,M6: B,N3: B] :
          ( order_mono(A,B,F2)
         => ( ( image(A,B,F2,set_or7035219750837199246ssThan(A,M2,N)) = set_or7035219750837199246ssThan(B,M6,N3) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),N))
             => ( aa(A,B,F2,M2) = M6 ) ) ) ) ) ).

% mono_image_least
tff(fact_6254_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F2: fun(A,A),P2: A,K: nat] :
          ( order_mono(A,A,F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,F2,P2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_6255_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F2: fun(A,A),P2: A,K: nat] :
          ( order_mono(A,A,F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,P2)),P2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_6256_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),I: nat,J2: nat,X: A,Y2: A] :
          ( order_mono(A,A,F2)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F2,X)))
               => pp(aa(A,bool,aa(A,fun(A,bool),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),X)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J2),F2),Y2))) ) ) ) ) ) ).

% funpow_mono2
tff(fact_6257_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_antimono(nat,A,aTP_Lamp_zw(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_6258_eventually__filtercomap__at__top__dense,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(B,bool),F2: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F2,at_top(A)))
        <=> ? [N7: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N7),aa(B,A,F2,X4)))
             => pp(aa(B,bool,P,X4)) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_6259_eventually__filtercomap__at__bot__dense,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(B,bool),F2: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F2,at_bot(A)))
        <=> ? [N7: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X4)),N7))
             => pp(aa(B,bool,P,X4)) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_6260_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M2: nat,N: nat,F2: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( order_mono(A,A,F2)
           => pp(aa(A,bool,aa(A,fun(A,bool),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)),N),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)),M2),F2),top_top(A)))) ) ) ) ).

% funpow_increasing
tff(fact_6261_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M2: nat,N: nat,F2: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( order_mono(A,A,F2)
           => pp(aa(A,bool,aa(A,fun(A,bool),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)),M2),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)),N),F2),bot_bot(A)))) ) ) ) ).

% funpow_decreasing
tff(fact_6262_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => order_mono(nat,nat,aTP_Lamp_zx(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_6263_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( pp(aa(set(A),bool,finite_finite(A),image(nat,A,F2,top_top(set(nat)))))
         => ( order_mono(nat,A,F2)
           => ( ! [N2: nat] :
                  ( ( aa(nat,A,F2,N2) = aa(nat,A,F2,aa(nat,nat,suc,N2)) )
                 => ( aa(nat,A,F2,aa(nat,nat,suc,N2)) = aa(nat,A,F2,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) )
             => ? [N9: nat] :
                  ( ! [N4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N9))
                     => ! [M: nat] :
                          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N9))
                         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N4))
                           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,M)),aa(nat,A,F2,N4))) ) ) )
                  & ! [N4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N4))
                     => ( aa(nat,A,F2,N9) = aa(nat,A,F2,N4) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_6264_tendsto__at__left__sequentially,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(A) )
     => ! [B2: B,A3: B,X6: fun(B,A),L4: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),A3))
         => ( ! [S7: fun(nat,B)] :
                ( ! [N4: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(nat,B,S7,N4)),A3))
               => ( ! [N4: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),aa(nat,B,S7,N4)))
                 => ( order_mono(nat,B,S7)
                   => ( filterlim(nat,B,S7,topolo7230453075368039082e_nhds(B,A3),at_top(nat))
                     => filterlim(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_zy(fun(B,A),fun(fun(nat,B),fun(nat,A)),X6),S7),topolo7230453075368039082e_nhds(A,L4),at_top(nat)) ) ) ) )
           => filterlim(B,A,X6,topolo7230453075368039082e_nhds(A,L4),topolo174197925503356063within(B,A3,set_ord_lessThan(B,A3))) ) ) ) ).

% tendsto_at_left_sequentially
tff(fact_6265_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys2 )
    <=> ? [F5: fun(nat,nat)] :
          ( order_mono(nat,nat,F5)
          & ( image(nat,nat,F5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys2)) )
          & ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys2),aa(nat,nat,F5,I2)) ) )
          & ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))) )
              <=> ( aa(nat,nat,F5,I2) = aa(nat,nat,F5,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_6266_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F2: fun(nat,set(A)),S3: set(A)] :
      ( ! [I3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F2,I3)),S3))
     => ( pp(aa(set(A),bool,finite_finite(A),S3))
       => ( ? [N8: nat] :
              ( ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N8))
                 => ! [M3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N8))
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N2))
                       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F2,M3)),aa(nat,set(A),F2,N2))) ) ) )
              & ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
                 => ( aa(nat,set(A),F2,N8) = aa(nat,set(A),F2,N2) ) ) )
         => ( 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_6267_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,X,Y2)) = Y2 ) ) ) ).

% Sup_atLeastLessThan
tff(fact_6268_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y2,X)) = X ) ) ) ).

% cSup_atLeastLessThan
tff(fact_6269_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,X,Y2)) = Y2 ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_6270_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y2,X)) = X ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_6271_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,X,Y2)) = Y2 ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_6272_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y2,X)) = X ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_6273_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),top_top(A)))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_ord_greaterThan(A,X)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_6274_remdups__adj__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6275_finite__subset__Union,axiom,
    ! [A: $tType,A4: set(A),B10: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B10)))
       => ~ ! [F7: set(set(A))] :
              ( pp(aa(set(set(A)),bool,finite_finite(set(A)),F7))
             => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F7),B10))
               => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F7))) ) ) ) ) ).

% finite_subset_Union
tff(fact_6276_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),B5: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,B5))),A3) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_zz(fun(B,A),fun(A,fun(B,A)),F2),A3),B5)) ) ).

% SUP_inf
tff(fact_6277_Sup__inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B5: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B5)),A3) = aa(set(A),A,complete_Sup_Sup(A),image(A,A,aTP_Lamp_aaa(A,fun(A,A),A3),B5)) ) ).

% Sup_inf
tff(fact_6278_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F2: fun(B,A),B5: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,B5))) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aab(A,fun(fun(B,A),fun(B,A)),A3),F2),B5)) ) ).

% inf_SUP
tff(fact_6279_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),A4: set(B),G: fun(C,A),B5: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))),aa(set(A),A,complete_Sup_Sup(A),image(C,A,G,B5))) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_aad(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B5),A4)) ) ).

% SUP_inf_distrib2
tff(fact_6280_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A3: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( ! [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),A3)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A3)) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_6281_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B5: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B5)),A3) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B5))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),A3) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_6282_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y2: A,X6: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(set(A),A,complete_Sup_Sup(A),X6)))
         => ( ( X6 != bot_bot(set(A)) )
           => ~ ! [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X2)) ) ) ) ) ).

% less_cSupE
tff(fact_6283_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X6)))
           => ? [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X2)) ) ) ) ) ).

% less_cSupD
tff(fact_6284_inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B5: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),B5)) = aa(set(A),A,complete_Sup_Sup(A),image(A,A,aa(A,fun(A,A),inf_inf(A),A3),B5)) ) ).

% inf_Sup
tff(fact_6285_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A4: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,A4)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),X)) )
           => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Sup_Sup(A),A4)),A4)) ) ) ) ).

% Sup_notin_open
tff(fact_6286_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A3))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A3)) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_6287_sum_OUnion__comp,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B5: set(set(B)),G: fun(B,A)] :
          ( ! [X2: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X2),B5))
             => pp(aa(set(B),bool,finite_finite(B),X2)) )
         => ( ! [A12: set(B)] :
                ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A12),B5))
               => ! [A23: set(B)] :
                    ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A23),B5))
                   => ( ( A12 != A23 )
                     => ! [X2: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A12))
                         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A23))
                           => ( aa(B,A,G,X2) = zero_zero(A) ) ) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B5)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7311177749621191930dd_sum(set(B),A)),groups7311177749621191930dd_sum(B,A)),G),B5) ) ) ) ) ).

% sum.Union_comp
tff(fact_6288_prod_OUnion__comp,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B5: set(set(B)),G: fun(B,A)] :
          ( ! [X2: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X2),B5))
             => pp(aa(set(B),bool,finite_finite(B),X2)) )
         => ( ! [A12: set(B)] :
                ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A12),B5))
               => ! [A23: set(B)] :
                    ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A23),B5))
                   => ( ( A12 != A23 )
                     => ! [X2: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A12))
                         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A23))
                           => ( aa(B,A,G,X2) = one_one(A) ) ) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B5)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7121269368397514597t_prod(set(B),A)),groups7121269368397514597t_prod(B,A)),G),B5) ) ) ) ) ).

% prod.Union_comp
tff(fact_6289_UN__UN__finite__eq,axiom,
    ! [A: $tType,A4: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),aTP_Lamp_aae(fun(nat,set(A)),fun(nat,set(A)),A4),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A4,top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_6290_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),L))),E2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),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))),E2)) ) ) ) ).

% cSup_asclose
tff(fact_6291_remdups__adj__adjacent,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6292_UN__finite__subset,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),C5: set(A)] :
      ( ! [N2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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),N2)))),C5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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))))),C5)) ) ).

% UN_finite_subset
tff(fact_6293_UN__finite2__eq,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),B5: fun(nat,set(A)),K: nat] :
      ( ! [N2: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A4,set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),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),B5,top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_6294_UN__finite2__subset,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),B5: fun(nat,set(A)),K: nat] :
      ( ! [N2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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),N2)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))))))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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),B5,top_top(set(nat)))))) ) ).

% UN_finite2_subset
tff(fact_6295_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) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),top_top(A)))
             => ? [Xa3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa3),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),aa(B,A,F2,Xa3))) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_6296_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A4) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),top_top(A)))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Xa3)) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_6297_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_6298_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,S3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),S3)))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X4)) ) ) ) ).

% less_Sup_iff
tff(fact_6299_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,A4: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A4)))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X4)) ) ) ) ) ).

% le_Sup_iff
tff(fact_6300_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,F2: fun(B,A),A4: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))))
        <=> ? [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F2,X4))) ) ) ) ).

% less_SUP_iff
tff(fact_6301_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A4: set(B),Y2: A,I: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))),Y2))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,I)),Y2)) ) ) ) ).

% SUP_lessD
tff(fact_6302_UN__extend__simps_I6_J,axiom,
    ! [L5: $tType,K8: $tType,A4: fun(K8,set(L5)),C5: set(K8),B5: set(L5)] : aa(set(L5),set(L5),aa(set(L5),fun(set(L5),set(L5)),minus_minus(set(L5)),aa(set(set(L5)),set(L5),complete_Sup_Sup(set(L5)),image(K8,set(L5),A4,C5))),B5) = aa(set(set(L5)),set(L5),complete_Sup_Sup(set(L5)),image(K8,set(L5),aa(set(L5),fun(K8,set(L5)),aTP_Lamp_aaf(fun(K8,set(L5)),fun(set(L5),fun(K8,set(L5))),A4),B5),C5)) ).

% UN_extend_simps(6)
tff(fact_6303_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,F2: fun(B,A),A4: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),aa(B,A,F2,X4))) ) ) ) ) ).

% le_SUP_iff
tff(fact_6304_card__UNION,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite(set(A)),A4))
     => ( ! [X2: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X2),A4))
           => pp(aa(set(A),bool,finite_finite(A),X2)) )
       => ( 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,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_aag(set(set(A)),int)),collect(set(set(A)),aTP_Lamp_aah(set(set(A)),fun(set(set(A)),bool),A4)))) ) ) ) ).

% card_UNION
tff(fact_6305_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_6306_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_6307_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A4) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa3),X4)) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_6308_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,X,Y2)) = X ) ) ) ).

% Inf_atLeastLessThan
tff(fact_6309_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y2,X)) = Y2 ) ) ) ).

% cInf_atLeastLessThan
tff(fact_6310_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,X,Y2)) = X ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_6311_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y2,X)) = Y2 ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_6312_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,X,Y2)) = X ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_6313_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y2,X)) = Y2 ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_6314_length__remdups__leq,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6315_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) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
             => ? [Xa3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa3),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,Xa3)),X4)) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_6316_Compl__INT,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A4: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),B5,A4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),aTP_Lamp_aai(fun(B,set(A)),fun(B,set(A)),B5),A4)) ).

% Compl_INT
tff(fact_6317_Compl__UN,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A4: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),B5,A4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),aTP_Lamp_aai(fun(B,set(A)),fun(B,set(A)),B5),A4)) ).

% Compl_UN
tff(fact_6318_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_ord_lessThan(A,X)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_6319_INT__simps_I3_J,axiom,
    ! [E5: $tType,F: $tType,C5: set(E5),A4: fun(E5,set(F)),B5: set(F)] :
      ( ( ( C5 = bot_bot(set(E5)) )
       => ( aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),image(E5,set(F),aa(set(F),fun(E5,set(F)),aTP_Lamp_aaj(fun(E5,set(F)),fun(set(F),fun(E5,set(F))),A4),B5),C5)) = top_top(set(F)) ) )
      & ( ( C5 != bot_bot(set(E5)) )
       => ( aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),image(E5,set(F),aa(set(F),fun(E5,set(F)),aTP_Lamp_aaj(fun(E5,set(F)),fun(set(F),fun(E5,set(F))),A4),B5),C5)) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),image(E5,set(F),A4,C5))),B5) ) ) ) ).

% INT_simps(3)
tff(fact_6320_INT__simps_I4_J,axiom,
    ! [G4: $tType,H6: $tType,C5: set(H6),A4: set(G4),B5: fun(H6,set(G4))] :
      ( ( ( C5 = bot_bot(set(H6)) )
       => ( aa(set(set(G4)),set(G4),complete_Inf_Inf(set(G4)),image(H6,set(G4),aa(fun(H6,set(G4)),fun(H6,set(G4)),aTP_Lamp_aak(set(G4),fun(fun(H6,set(G4)),fun(H6,set(G4))),A4),B5),C5)) = top_top(set(G4)) ) )
      & ( ( C5 != bot_bot(set(H6)) )
       => ( aa(set(set(G4)),set(G4),complete_Inf_Inf(set(G4)),image(H6,set(G4),aa(fun(H6,set(G4)),fun(H6,set(G4)),aTP_Lamp_aak(set(G4),fun(fun(H6,set(G4)),fun(H6,set(G4))),A4),B5),C5)) = aa(set(G4),set(G4),aa(set(G4),fun(set(G4),set(G4)),minus_minus(set(G4)),A4),aa(set(set(G4)),set(G4),complete_Sup_Sup(set(G4)),image(H6,set(G4),B5,C5))) ) ) ) ).

% INT_simps(4)
tff(fact_6321_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y2: A,F2: fun(B,A),A4: set(B),I: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(B,A,F2,I))) ) ) ) ).

% less_INF_D
tff(fact_6322_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B),A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))),A3))
        <=> ? [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X4)),A3)) ) ) ) ).

% INF_less_iff
tff(fact_6323_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S3: set(A),A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S3)),A3))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A3)) ) ) ) ).

% Inf_less_iff
tff(fact_6324_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A3: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( ! [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X2)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_6325_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),X))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4)) ) ) ) ) ).

% Inf_le_iff
tff(fact_6326_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Z))
           => ? [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z)) ) ) ) ) ).

% cInf_lessD
tff(fact_6327_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A4: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,A4)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X2)) )
           => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Inf_Inf(A),A4)),A4)) ) ) ) ).

% Inf_notin_open
tff(fact_6328_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))),X))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X4)),Y4)) ) ) ) ) ).

% INF_le_iff
tff(fact_6329_Sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A4: set(set(A))] : aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),A4)) = aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),collect(set(A),aTP_Lamp_aal(set(set(A)),fun(set(A),bool),A4)))) ) ).

% Sup_Inf
tff(fact_6330_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X4)) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_6331_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),A3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),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_6332_uminus__Inf,axiom,
    ! [A: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [A4: set(A)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),A4)) = aa(set(A),A,complete_Sup_Sup(A),image(A,A,uminus_uminus(A),A4)) ) ).

% uminus_Inf
tff(fact_6333_uminus__Sup,axiom,
    ! [A: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [A4: set(A)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),A4)) = aa(set(A),A,complete_Inf_Inf(A),image(A,A,uminus_uminus(A),A4)) ) ).

% uminus_Sup
tff(fact_6334_uminus__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B5: fun(B,A),A4: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,B5,A4))) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aTP_Lamp_aam(fun(B,A),fun(B,A),B5),A4)) ) ).

% uminus_INF
tff(fact_6335_uminus__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B5: fun(B,A),A4: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,B5,A4))) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aTP_Lamp_aam(fun(B,A),fun(B,A),B5),A4)) ) ).

% uminus_SUP
tff(fact_6336_UN__extend__simps_I7_J,axiom,
    ! [M11: $tType,N10: $tType,A4: set(M11),B5: fun(N10,set(M11)),C5: set(N10)] : aa(set(M11),set(M11),aa(set(M11),fun(set(M11),set(M11)),minus_minus(set(M11)),A4),aa(set(set(M11)),set(M11),complete_Inf_Inf(set(M11)),image(N10,set(M11),B5,C5))) = aa(set(set(M11)),set(M11),complete_Sup_Sup(set(M11)),image(N10,set(M11),aa(fun(N10,set(M11)),fun(N10,set(M11)),aTP_Lamp_aan(set(M11),fun(fun(N10,set(M11)),fun(N10,set(M11))),A4),B5),C5)) ).

% UN_extend_simps(7)
tff(fact_6337_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_6338_SUP__INF__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: fun(B,A),A4: set(set(B))] : aa(set(A),A,complete_Sup_Sup(A),image(set(B),A,aTP_Lamp_aao(fun(B,A),fun(set(B),A),G),A4)) = aa(set(A),A,complete_Inf_Inf(A),image(set(B),A,aTP_Lamp_aap(fun(B,A),fun(set(B),A),G),collect(set(B),aTP_Lamp_aaq(set(set(B)),fun(set(B),bool),A4)))) ) ).

% SUP_INF_set
tff(fact_6339_INF__SUP__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: fun(B,A),A4: set(set(B))] : aa(set(A),A,complete_Inf_Inf(A),image(set(B),A,aTP_Lamp_aap(fun(B,A),fun(set(B),A),G),A4)) = aa(set(A),A,complete_Sup_Sup(A),image(set(B),A,aTP_Lamp_aao(fun(B,A),fun(set(B),A),G),collect(set(B),aTP_Lamp_aaq(set(set(B)),fun(set(B),bool),A4)))) ) ).

% INF_SUP_set
tff(fact_6340_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),L))),E2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),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))),E2)) ) ) ) ).

% cInf_asclose
tff(fact_6341_mono__bij__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & comple5582772986160207858norder(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( order_mono(A,B,F2)
         => ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(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_bij_Inf
tff(fact_6342_INT__extend__simps_I3_J,axiom,
    ! [F: $tType,E5: $tType,C5: set(E5),A4: fun(E5,set(F)),B5: set(F)] :
      ( ( ( C5 = bot_bot(set(E5)) )
       => ( aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),image(E5,set(F),A4,C5))),B5) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),top_top(set(F))),B5) ) )
      & ( ( C5 != bot_bot(set(E5)) )
       => ( aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),image(E5,set(F),A4,C5))),B5) = aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),image(E5,set(F),aa(set(F),fun(E5,set(F)),aTP_Lamp_aaj(fun(E5,set(F)),fun(set(F),fun(E5,set(F))),A4),B5),C5)) ) ) ) ).

% INT_extend_simps(3)
tff(fact_6343_INT__extend__simps_I4_J,axiom,
    ! [G4: $tType,H6: $tType,C5: set(H6),A4: set(G4),B5: fun(H6,set(G4))] :
      ( ( ( C5 = bot_bot(set(H6)) )
       => ( aa(set(G4),set(G4),aa(set(G4),fun(set(G4),set(G4)),minus_minus(set(G4)),A4),aa(set(set(G4)),set(G4),complete_Sup_Sup(set(G4)),image(H6,set(G4),B5,C5))) = A4 ) )
      & ( ( C5 != bot_bot(set(H6)) )
       => ( aa(set(G4),set(G4),aa(set(G4),fun(set(G4),set(G4)),minus_minus(set(G4)),A4),aa(set(set(G4)),set(G4),complete_Sup_Sup(set(G4)),image(H6,set(G4),B5,C5))) = aa(set(set(G4)),set(G4),complete_Inf_Inf(set(G4)),image(H6,set(G4),aa(fun(H6,set(G4)),fun(H6,set(G4)),aTP_Lamp_aak(set(G4),fun(fun(H6,set(G4)),fun(H6,set(G4))),A4),B5),C5)) ) ) ) ).

% INT_extend_simps(4)
tff(fact_6344_INF__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Inf_Inf(A),image(B,A,aTP_Lamp_aas(fun(C,fun(B,A)),fun(B,A),P),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),image(fun(B,C),A,aTP_Lamp_aau(fun(C,fun(B,A)),fun(fun(B,C),A),P),top_top(set(fun(B,C))))) ) ).

% INF_SUP
tff(fact_6345_SUP__INF,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),image(B,A,aTP_Lamp_aav(fun(C,fun(B,A)),fun(B,A),P),top_top(set(B)))) = aa(set(A),A,complete_Inf_Inf(A),image(fun(B,C),A,aTP_Lamp_aaw(fun(C,fun(B,A)),fun(fun(B,C),A),P),top_top(set(fun(B,C))))) ) ).

% SUP_INF
tff(fact_6346_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: A,B5: 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_aax(A,fun(nat,A),B5),collect(nat,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B5) ) ).

% INF_nat_binary
tff(fact_6347_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => inj_on(real,real,aTP_Lamp_oy(nat,fun(real,real),N),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_6348_inj__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A4: set(A)] : inj_on(A,A,uminus_uminus(A),A4) ) ).

% inj_uminus
tff(fact_6349_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_6350_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_aay(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6351_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: A,A4: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),insert(A,A3),A4))
    <=> ( inj_on(A,B,F2,A4)
        & ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A3)),image(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))))))) ) ) ).

% inj_on_insert
tff(fact_6352_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S3: set(A),F2: fun(B,A),T5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S3),image(B,A,F2,T5)))
    <=> ? [U5: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),U5),T5))
          & inj_on(B,A,F2,U5)
          & ( S3 = image(B,A,F2,U5) ) ) ) ).

% subset_image_inj
tff(fact_6353_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_6354_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : inj_on(A,A,aTP_Lamp_mj(A,fun(A,A),A3),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_6355_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_6356_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X2: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3))
             => ( aa(A,B,F2,X2) != aa(A,B,F2,Y3) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_6357_inj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),N: 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)),N),F2),top_top(set(A))) ) ).

% inj_fn
tff(fact_6358_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_6359_inj__on__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B5: set(A)] :
      ( inj_on(A,B,F2,A4)
     => inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) ) ).

% inj_on_diff
tff(fact_6360_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X2: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A4))
                 => ( aa(A,B,F2,X2) != aa(A,B,F2,Y3) ) ) ) )
         => ( ! [X2: A,Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A4))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y3))
                    | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X2)) ) ) )
           => inj_on(A,B,F2,A4) ) ) ) ).

% linorder_inj_onI
tff(fact_6361_Inf__int__def,axiom,
    ! [X6: set(int)] : aa(set(int),int,complete_Inf_Inf(int),X6) = aa(int,int,uminus_uminus(int),aa(set(int),int,complete_Sup_Sup(int),image(int,int,uminus_uminus(int),X6))) ).

% Inf_int_def
tff(fact_6362_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A4: set(A),A13: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ? [F5: fun(A,B)] :
            ( inj_on(A,B,F5,A4)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,F5,A4)),A13)) )
      <=> ? [G3: fun(B,A)] : image(B,A,G3,A13) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_6363_image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B5: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( image(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),image(A,B,F2,A4)),image(A,B,F2,B5)) ) ) ).

% image_set_diff
tff(fact_6364_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C5: set(A),A4: set(A),B5: set(A)] :
      ( inj_on(A,B,F2,C5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),C5))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C5))
         => ( image(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),image(A,B,F2,A4)),image(A,B,F2,B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_6365_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6366_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A3: A,X: A,B2: A,F2: fun(A,B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B2),F2)
             => ( inj_on(A,B,F2,set_or1337092689740270186AtMost(A,A3,B2))
               => ( ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,A3)),aa(A,B,F2,X)))
                    & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,B2))) )
                  | ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,B2)),aa(A,B,F2,X)))
                    & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,A3))) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_6367_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),A4))),aa(set(B),set(B),uminus_uminus(set(B)),image(A,B,F2,A4)))) ) ).

% inj_image_Compl_subset
tff(fact_6368_log__inj,axiom,
    ! [B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => inj_on(real,real,log2(B2),set_ord_greaterThan(real,zero_zero(real))) ) ).

% log_inj
tff(fact_6369_funpow__inj__finite,axiom,
    ! [A: $tType,P2: fun(A,A),X: A] :
      ( inj_on(A,A,P2,top_top(set(A)))
     => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_aaz(fun(A,A),fun(A,fun(A,bool)),P2),X))))
       => ~ ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),P2),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_6370_at__within__order,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,S: set(A)] :
          ( ( top_top(set(A)) != aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
         => ( topolo174197925503356063within(A,X,S) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aba(A,fun(set(A),fun(A,filter(A))),X),S),set_ord_greaterThan(A,X)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_abb(A,fun(set(A),fun(A,filter(A))),X),S),set_ord_lessThan(A,X)))) ) ) ) ).

% at_within_order
tff(fact_6371_surj__int__decode,axiom,
    image(nat,int,nat_int_decode,top_top(set(nat))) = top_top(set(int)) ).

% surj_int_decode
tff(fact_6372_int__encode__inverse,axiom,
    ! [X: int] : aa(nat,int,nat_int_decode,aa(int,nat,nat_int_encode,X)) = X ).

% int_encode_inverse
tff(fact_6373_int__decode__inverse,axiom,
    ! [N: nat] : aa(int,nat,nat_int_encode,aa(nat,int,nat_int_decode,N)) = N ).

% int_decode_inverse
tff(fact_6374_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_6375_inj__Some,axiom,
    ! [A: $tType,A4: set(A)] : inj_on(A,option(A),some(A),A4) ).

% inj_Some
tff(fact_6376_inj__prod__encode,axiom,
    ! [A4: set(product_prod(nat,nat))] : inj_on(product_prod(nat,nat),nat,nat_prod_encode,A4) ).

% inj_prod_encode
tff(fact_6377_inj__setminus,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A4: set(set(A))] : inj_on(set(A),set(A),uminus_uminus(set(A)),A4) ) ).

% inj_setminus
tff(fact_6378_inj__Suc,axiom,
    ! [N6: set(nat)] : inj_on(nat,nat,suc,N6) ).

% inj_Suc
tff(fact_6379_inj__int__decode,axiom,
    ! [A4: set(nat)] : inj_on(nat,int,nat_int_decode,A4) ).

% inj_int_decode
tff(fact_6380_inj__on__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N6: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N6) ) ).

% inj_on_of_nat
tff(fact_6381_inj__int__encode,axiom,
    ! [A4: set(int)] : inj_on(int,nat,nat_int_encode,A4) ).

% inj_int_encode
tff(fact_6382_inj__on__diff__nat,axiom,
    ! [N6: set(nat),K: nat] :
      ( ! [N2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2)) )
     => inj_on(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),K),N6) ) ).

% inj_on_diff_nat
tff(fact_6383_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,collect(set(nat),finite_finite(nat))) ).

% inj_on_set_encode
tff(fact_6384_int__decode__eq,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa(nat,int,nat_int_decode,X) = aa(nat,int,nat_int_decode,Y2) )
    <=> ( X = Y2 ) ) ).

% int_decode_eq
tff(fact_6385_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ? [N2: nat,F3: fun(nat,A)] :
          ( ( A4 = image(nat,A,F3,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N2))) )
          & inj_on(nat,A,F3,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N2))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6386_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ? [F3: fun(A,nat),N2: nat] :
          ( ( image(A,nat,F3,A4) = collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N2)) )
          & inj_on(A,nat,F3,A4) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6387_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X2: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),I5))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs))) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_6388_infinite__countable__subset,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),S3))
     => ? [F3: fun(nat,A)] :
          ( inj_on(nat,A,F3,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(nat,A,F3,top_top(set(nat)))),S3)) ) ) ).

% infinite_countable_subset
tff(fact_6389_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),S3))
    <=> ? [F5: fun(nat,A)] :
          ( inj_on(nat,A,F5,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(nat,A,F5,top_top(set(nat)))),S3)) ) ) ).

% infinite_iff_countable_subset
tff(fact_6390_inj__on__funpow__least,axiom,
    ! [A: $tType,N: 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)),N),F2),S) = S )
     => ( ! [M3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M3),F2),S) != S ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_abc(fun(A,A),fun(A,fun(nat,A)),F2),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_6391_bij__int__decode,axiom,
    bij_betw(nat,int,nat_int_decode,top_top(set(nat)),top_top(set(int))) ).

% bij_int_decode
tff(fact_6392_at__left__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
         => ( topolo174197925503356063within(A,X,set_ord_lessThan(A,X)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(A,filter(A),aTP_Lamp_abd(A,fun(A,filter(A)),X),set_ord_lessThan(A,X))) ) ) ) ).

% at_left_eq
tff(fact_6393_at__right__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( topolo174197925503356063within(A,X,set_ord_greaterThan(A,X)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(A,filter(A),aTP_Lamp_abe(A,fun(A,filter(A)),X),set_ord_greaterThan(A,X))) ) ) ) ).

% at_right_eq
tff(fact_6394_at__within__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A,S: set(A)] : topolo174197925503356063within(A,A3,S) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A3)),principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) ) ).

% at_within_def
tff(fact_6395_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A,S: set(A)] : topolo174197925503356063within(A,X,S) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_abf(A,fun(set(A),fun(set(A),filter(A))),X),S),collect(set(A),aTP_Lamp_abg(A,fun(set(A),bool),X)))) ) ).

% at_within_eq
tff(fact_6396_has__derivative__power__int_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,N: int,S3: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,aTP_Lamp_abh(int,fun(A,A),N),aa(int,fun(A,A),aTP_Lamp_abi(A,fun(int,fun(A,A)),X),N),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_power_int'
tff(fact_6397_has__derivative__power__int,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),X: C,F6: fun(C,A),S3: set(C),N: int] :
          ( ( aa(C,A,F2,X) != zero_zero(A) )
         => ( has_derivative(C,A,F2,F6,topolo174197925503356063within(C,X,S3))
           => has_derivative(C,A,aa(int,fun(C,A),aTP_Lamp_abj(fun(C,A),fun(int,fun(C,A)),F2),N),aa(int,fun(C,A),aa(fun(C,A),fun(int,fun(C,A)),aa(C,fun(fun(C,A),fun(int,fun(C,A))),aTP_Lamp_abk(fun(C,A),fun(C,fun(fun(C,A),fun(int,fun(C,A)))),F2),X),F6),N),topolo174197925503356063within(C,X,S3)) ) ) ) ).

% has_derivative_power_int
tff(fact_6398_power__int__1__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int] : power_int(A,one_one(A),N) = one_one(A) ) ).

% power_int_1_left
tff(fact_6399_power__int__1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y2: A] : power_int(A,Y2,one_one(int)) = Y2 ) ).

% power_int_1_right
tff(fact_6400_power__int__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] : aa(A,A,sgn_sgn(A),power_int(A,A3,N)) = power_int(A,aa(A,A,sgn_sgn(A),A3),N) ) ).

% power_int_sgn
tff(fact_6401_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,W: num,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W)),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),power_int(A,aa(num,A,numeral_numeral(A),W),M2)) ) ).

% power_int_mult_distrib_numeral2
tff(fact_6402_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W: num,Y2: A,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y2),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W),M2)),power_int(A,Y2,M2)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_6403_power__int__0__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: int] :
          ( ( M2 != zero_zero(int) )
         => ( power_int(A,zero_zero(A),M2) = zero_zero(A) ) ) ) ).

% power_int_0_left
tff(fact_6404_power__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] :
          ( ( power_int(A,X,N) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( N != zero_zero(int) ) ) ) ) ).

% power_int_eq_0_iff
tff(fact_6405_power__int__0__right,axiom,
    ! [B: $tType] :
      ( ( inverse(B)
        & power(B) )
     => ! [X: B] : power_int(B,X,zero_zero(int)) = one_one(B) ) ).

% power_int_0_right
tff(fact_6406_abs__power__int__minus,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] : aa(A,A,abs_abs(A),power_int(A,aa(A,A,uminus_uminus(A),A3),N)) = aa(A,A,abs_abs(A),power_int(A,A3,N)) ) ).

% abs_power_int_minus
tff(fact_6407_power__int__of__nat,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [X: A,N: nat] : power_int(A,X,aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N) ) ).

% power_int_of_nat
tff(fact_6408_power__int__mult__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: num,N: num] : power_int(A,power_int(A,X,aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ).

% power_int_mult_numeral
tff(fact_6409_power__int__minus__one__mult__self_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: 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)),M2)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),B2)) = B2 ) ).

% power_int_minus_one_mult_self'
tff(fact_6410_power__int__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)) = one_one(A) ) ).

% power_int_minus_one_mult_self
tff(fact_6411_power__int__numeral,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [X: A,N: num] : power_int(A,X,aa(num,int,numeral_numeral(int),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),N)) ) ).

% power_int_numeral
tff(fact_6412_power__int__minus1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y2: A] : power_int(A,Y2,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Y2) ) ).

% power_int_minus1_right
tff(fact_6413_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M2))),power_int(A,X,aa(num,int,numeral_numeral(int),N))) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ).

% power_int_add_numeral
tff(fact_6414_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M2))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)))),B2) ) ).

% power_int_add_numeral2
tff(fact_6415_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,B2,N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_6416_power__int__minus__left__odd,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int,A3: A] :
          ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
         => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,uminus_uminus(A),power_int(A,A3,N)) ) ) ) ).

% power_int_minus_left_odd
tff(fact_6417_power__int__minus__left__even,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
         => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = power_int(A,A3,N) ) ) ) ).

% power_int_minus_left_even
tff(fact_6418_power__int__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : power_int(A,X,aa(int,int,uminus_uminus(int),N)) = aa(A,A,inverse_inverse(A),power_int(A,X,N)) ) ).

% power_int_minus
tff(fact_6419_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_less_power_int
tff(fact_6420_power__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] : aa(A,A,abs_abs(A),power_int(A,A3,N)) = power_int(A,aa(A,A,abs_abs(A),A3),N) ) ).

% power_int_abs
tff(fact_6421_power__int__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : power_int(A,aa(A,A,inverse_inverse(A),X),N) = aa(A,A,inverse_inverse(A),power_int(A,X,N)) ) ).

% power_int_inverse
tff(fact_6422_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y2: A,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y2),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),power_int(A,Y2,M2)) ) ).

% power_int_mult_distrib
tff(fact_6423_power__int__commutes,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,N)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,N)) ) ).

% power_int_commutes
tff(fact_6424_power__int__mult,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: int,N: int] : power_int(A,X,aa(int,int,aa(int,fun(int,int),times_times(int),M2),N)) = power_int(A,power_int(A,X,M2),N) ) ).

% power_int_mult
tff(fact_6425_power__int__divide__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y2: A,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y2),M2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,X,M2)),power_int(A,Y2,M2)) ) ).

% power_int_divide_distrib
tff(fact_6426_power__int__not__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( N = zero_zero(int) ) )
         => ( power_int(A,X,N) != zero_zero(A) ) ) ) ).

% power_int_not_zero
tff(fact_6427_power__int__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),X),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),power_int(A,X,N)) ) ).

% power_int_one_over
tff(fact_6428_continuous__on__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),N: int] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
               => ( aa(A,B,F2,X2) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aa(int,fun(A,B),aTP_Lamp_abl(fun(A,B),fun(int,fun(A,B)),F2),N)) ) ) ) ).

% continuous_on_power_int
tff(fact_6429_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_le_power_int
tff(fact_6430_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: int] :
          ( ( ( M2 = zero_zero(int) )
           => ( power_int(A,zero_zero(A),M2) = one_one(A) ) )
          & ( ( M2 != zero_zero(int) )
           => ( power_int(A,zero_zero(A),M2) = zero_zero(A) ) ) ) ) ).

% power_int_0_left_If
tff(fact_6431_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N6: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,A3,N6))) ) ) ) ).

% power_int_increasing
tff(fact_6432_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N6: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N)),power_int(A,A3,N6))) ) ) ) ).

% power_int_strict_increasing
tff(fact_6433_power__int__minus__one__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,uminus_uminus(int),N)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N) ) ).

% power_int_minus_one_minus
tff(fact_6434_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,M2: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( M2 != N ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),M2),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,X,M2)),power_int(A,X,N)) ) ) ) ).

% power_int_diff
tff(fact_6435_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_6436_tendsto__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A3: A,F4: filter(B),N: int] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( A3 != zero_zero(A) )
           => filterlim(B,A,aa(int,fun(B,A),aTP_Lamp_abm(fun(B,A),fun(int,fun(B,A)),F2),N),topolo7230453075368039082e_nhds(A,power_int(A,A3,N)),F4) ) ) ) ).

% tendsto_power_int
tff(fact_6437_continuous__at__within__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [A3: A,S: set(A),F2: fun(A,B),N: int] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),aa(int,fun(A,B),aTP_Lamp_abn(fun(A,B),fun(int,fun(A,B)),F2),N)) ) ) ) ).

% continuous_at_within_power_int
tff(fact_6438_differentiable__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S: set(A),N: int] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,B,F2,X) != zero_zero(B) )
           => differentiable(A,B,aa(int,fun(A,B),aTP_Lamp_abo(fun(A,B),fun(int,fun(A,B)),F2),N),topolo174197925503356063within(A,X,S)) ) ) ) ).

% differentiable_power_int
tff(fact_6439_continuous__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),N: int] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aa(int,fun(A,B),aTP_Lamp_abn(fun(A,B),fun(int,fun(A,B)),F2),N)) ) ) ) ).

% continuous_power_int
tff(fact_6440_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N6: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N6)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_6441_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),power_int(A,Y2,N))) ) ) ) ) ).

% power_int_mono
tff(fact_6442_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_6443_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),power_int(A,X,N))) ) ) ) ).

% one_le_power_int
tff(fact_6444_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),power_int(A,A3,N))) ) ) ) ).

% one_less_power_int
tff(fact_6445_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N) != zero_zero(int) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),power_int(A,X,N)) ) ) ) ).

% power_int_add
tff(fact_6446_power__int__minus__left__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( division_ring(A)
        & one(B)
        & uminus(B) )
     => ! [X: C,A3: A,N: int] :
          ( nO_MATCH(B,C,aa(B,B,uminus_uminus(B),one_one(B)),X)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N)),power_int(A,A3,N)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_6447_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N)),power_int(A,B2,N))) ) ) ) ) ).

% power_int_strict_mono
tff(fact_6448_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_antimono
tff(fact_6449_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),one_one(A))) ) ) ) ) ).

% power_int_le_one
tff(fact_6450_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N6: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
             => ( ( ( A3 != zero_zero(A) )
                  | ( N6 != zero_zero(int) )
                  | ( N = zero_zero(int) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N6)),power_int(A,A3,N))) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_6451_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M2: int,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,M2)),power_int(A,X,N)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M2),N)) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_6452_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M2: int,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,X,M2)),power_int(A,X,N)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M2),N)) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_6453_power__int__minus__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int,A3: A] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
           => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = power_int(A,A3,N) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
           => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,uminus_uminus(A),power_int(A,A3,N)) ) ) ) ) ).

% power_int_minus_left
tff(fact_6454_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( N != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int)))),X) = power_int(A,X,N) ) ) ) ).

% power_int_minus_mult
tff(fact_6455_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: int] :
          ( ( ( X != zero_zero(A) )
            | ( M2 != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,M2)) ) ) ) ).

% power_int_add_1'
tff(fact_6456_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: int] :
          ( ( ( X != zero_zero(A) )
            | ( M2 != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),X) ) ) ) ).

% power_int_add_1
tff(fact_6457_power__int__def,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [N: int,X: A] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( power_int(A,X,N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(int,nat,nat2,N)) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( power_int(A,X,N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),N))) ) ) ) ) ).

% power_int_def
tff(fact_6458_powr__real__of__int_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( ( X != zero_zero(real) )
          | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) )
       => ( aa(real,real,aa(real,fun(real,real),powr(real),X),aa(int,real,ring_1_of_int(real),N)) = power_int(real,X,N) ) ) ) ).

% powr_real_of_int'
tff(fact_6459_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S: set(A),N: int] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_abp(fun(A,A),fun(int,fun(A,A)),F2),N),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),N)),power_int(A,aa(A,A,F2,X),aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int))))),D2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_power_int
tff(fact_6460_power__int__numeral__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M2: num,N: num] : power_int(A,aa(num,A,numeral_numeral(A),M2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),pow(M2,N))) ) ).

% power_int_numeral_neg_numeral
tff(fact_6461_inverse__real_Otransfer,axiom,
    pp(aa(fun(real,real),bool,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(real,real),bool),bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real),aTP_Lamp_abs(fun(nat,rat),fun(nat,rat))),inverse_inverse(real))) ).

% inverse_real.transfer
tff(fact_6462_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one2) = X ).

% pow.simps(1)
tff(fact_6463_zero__real_Otransfer,axiom,
    pp(aa(real,bool,aa(fun(nat,rat),fun(real,bool),pcr_real,aTP_Lamp_abq(nat,rat)),zero_zero(real))) ).

% zero_real.transfer
tff(fact_6464_one__real_Otransfer,axiom,
    pp(aa(real,bool,aa(fun(nat,rat),fun(real,bool),pcr_real,aTP_Lamp_abt(nat,rat)),one_one(real))) ).

% one_real.transfer
tff(fact_6465_uminus__real_Otransfer,axiom,
    pp(aa(fun(real,real),bool,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(real,real),bool),bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real),aTP_Lamp_ov(fun(nat,rat),fun(nat,rat))),uminus_uminus(real))) ).

% uminus_real.transfer
tff(fact_6466_plus__real_Otransfer,axiom,
    pp(aa(fun(real,fun(real,real)),bool,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(real,fun(real,real)),bool),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_ow(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),plus_plus(real))) ).

% plus_real.transfer
tff(fact_6467_times__real_Otransfer,axiom,
    pp(aa(fun(real,fun(real,real)),bool,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(real,fun(real,real)),bool),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_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),times_times(real))) ).

% times_real.transfer
tff(fact_6468_Real_Opositive_Otransfer,axiom,
    pp(aa(fun(real,bool),bool,aa(fun(fun(nat,rat),bool),fun(fun(real,bool),bool),bNF_rel_fun(fun(nat,rat),real,bool,bool,pcr_real,fequal(bool)),aTP_Lamp_abu(fun(nat,rat),bool)),positive2)) ).

% Real.positive.transfer
tff(fact_6469_pow_Osimps_I3_J,axiom,
    ! [X: num,Y2: num] : pow(X,aa(num,num,bit1,Y2)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(X,Y2))),X) ).

% pow.simps(3)
tff(fact_6470_sqr__conv__mult,axiom,
    ! [X: num] : sqr(X) = aa(num,num,aa(num,fun(num,num),times_times(num),X),X) ).

% sqr_conv_mult
tff(fact_6471_Real_Opositive__mult,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,positive2,X))
     => ( pp(aa(real,bool,positive2,Y2))
       => pp(aa(real,bool,positive2,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y2))) ) ) ).

% Real.positive_mult
tff(fact_6472_Real_Opositive__zero,axiom,
    ~ pp(aa(real,bool,positive2,zero_zero(real))) ).

% Real.positive_zero
tff(fact_6473_Real_Opositive__add,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,positive2,X))
     => ( pp(aa(real,bool,positive2,Y2))
       => pp(aa(real,bool,positive2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y2))) ) ) ).

% Real.positive_add
tff(fact_6474_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_6475_sqr_Osimps_I2_J,axiom,
    ! [N: num] : sqr(aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,bit0,sqr(N))) ).

% sqr.simps(2)
tff(fact_6476_Real_Opositive__minus,axiom,
    ! [X: real] :
      ( ~ pp(aa(real,bool,positive2,X))
     => ( ( X != zero_zero(real) )
       => pp(aa(real,bool,positive2,aa(real,real,uminus_uminus(real),X))) ) ) ).

% Real.positive_minus
tff(fact_6477_less__real__def,axiom,
    ! [X: real,Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y2))
    <=> pp(aa(real,bool,positive2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Y2),X))) ) ).

% less_real_def
tff(fact_6478_pow_Osimps_I2_J,axiom,
    ! [X: num,Y2: num] : pow(X,aa(num,num,bit0,Y2)) = sqr(pow(X,Y2)) ).

% pow.simps(2)
tff(fact_6479_sqr_Osimps_I3_J,axiom,
    ! [N: num] : sqr(aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(N)),N))) ).

% sqr.simps(3)
tff(fact_6480_Real_Opositive_Orep__eq,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,positive2,X))
    <=> ? [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
          & ? [K3: nat] :
            ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,aa(real,fun(nat,rat),rep_real,X),N5))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_6481_Real_Opositive_Orsp,axiom,
    pp(aa(fun(fun(nat,rat),bool),bool,aa(fun(fun(nat,rat),bool),fun(fun(fun(nat,rat),bool),bool),bNF_rel_fun(fun(nat,rat),fun(nat,rat),bool,bool,realrel,fequal(bool)),aTP_Lamp_abu(fun(nat,rat),bool)),aTP_Lamp_abu(fun(nat,rat),bool))) ).

% Real.positive.rsp
tff(fact_6482_times__real_Orsp,axiom,
    pp(aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),bool,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),bool),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_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))))) ).

% times_real.rsp
tff(fact_6483_uminus__real_Orsp,axiom,
    pp(aa(fun(fun(nat,rat),fun(nat,rat)),bool,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(fun(nat,rat),fun(nat,rat)),bool),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel),aTP_Lamp_ov(fun(nat,rat),fun(nat,rat))),aTP_Lamp_ov(fun(nat,rat),fun(nat,rat)))) ).

% uminus_real.rsp
tff(fact_6484_plus__real_Orsp,axiom,
    pp(aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),bool,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),bool),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_ow(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),aTP_Lamp_ow(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))))) ).

% plus_real.rsp
tff(fact_6485_one__real_Orsp,axiom,
    pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,aTP_Lamp_abt(nat,rat)),aTP_Lamp_abt(nat,rat))) ).

% one_real.rsp
tff(fact_6486_zero__real_Orsp,axiom,
    pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,aTP_Lamp_abq(nat,rat)),aTP_Lamp_abq(nat,rat))) ).

% zero_real.rsp
tff(fact_6487_real_Orel__eq__transfer,axiom,
    pp(aa(fun(real,fun(real,bool)),bool,aa(fun(fun(nat,rat),fun(fun(nat,rat),bool)),fun(fun(real,fun(real,bool)),bool),bNF_rel_fun(fun(nat,rat),real,fun(fun(nat,rat),bool),fun(real,bool),pcr_real,bNF_rel_fun(fun(nat,rat),real,bool,bool,pcr_real,fequal(bool))),realrel),fequal(real))) ).

% real.rel_eq_transfer
tff(fact_6488_inverse__real_Orsp,axiom,
    pp(aa(fun(fun(nat,rat),fun(nat,rat)),bool,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(fun(nat,rat),fun(nat,rat)),bool),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel),aTP_Lamp_abs(fun(nat,rat),fun(nat,rat))),aTP_Lamp_abs(fun(nat,rat),fun(nat,rat)))) ).

% inverse_real.rsp
tff(fact_6489_Real_Opositive__def,axiom,
    positive2 = aa(fun(fun(nat,rat),bool),fun(real,bool),map_fun(real,fun(nat,rat),bool,bool,rep_real,id(bool)),aTP_Lamp_abu(fun(nat,rat),bool)) ).

% Real.positive_def
tff(fact_6490_UN__le__eq__Un0,axiom,
    ! [A: $tType,M10: fun(nat,set(A)),N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M10,set_ord_atMost(nat,N))) = 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),M10,set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M10,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_6491_sup_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),A3) = A3 ) ).

% sup.idem
tff(fact_6492_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),X) = X ) ).

% sup_idem
tff(fact_6493_sup_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).

% sup.left_idem
tff(fact_6494_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) ) ).

% sup_left_idem
tff(fact_6495_sup_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).

% sup.right_idem
tff(fact_6496_sup__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F2),G),X) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ).

% sup_apply
tff(fact_6497_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).

% sup.bounded_iff
tff(fact_6498_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Z)) ) ) ) ).

% le_sup_iff
tff(fact_6499_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).

% sup_top_right
tff(fact_6500_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).

% sup_top_left
tff(fact_6501_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),X) = X ) ).

% sup_bot_left
tff(fact_6502_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% sup_bot_right
tff(fact_6503_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y2: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) )
        <=> ( ( X = bot_bot(A) )
            & ( Y2 = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_6504_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) = bot_bot(A) )
        <=> ( ( X = bot_bot(A) )
            & ( Y2 = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_6505_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = bot_bot(A) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_6506_sup__bot_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A3) = A3 ) ).

% sup_bot.left_neutral
tff(fact_6507_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_6508_sup__bot_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),bot_bot(A)) = A3 ) ).

% sup_bot.right_neutral
tff(fact_6509_sup__inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)) = X ) ).

% sup_inf_absorb
tff(fact_6510_inf__sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)) = X ) ).

% inf_sup_absorb
tff(fact_6511_Un__Diff__cancel2,axiom,
    ! [A: $tType,B5: set(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A4) ).

% Un_Diff_cancel2
tff(fact_6512_Un__Diff__cancel,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5) ).

% Un_Diff_cancel
tff(fact_6513_sup__compl__top__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)) = top_top(A) ) ).

% sup_compl_top_left1
tff(fact_6514_sup__compl__top__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),Y2)) = top_top(A) ) ).

% sup_compl_top_left2
tff(fact_6515_boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),X) = top_top(A) ) ).

% boolean_algebra.disj_cancel_left
tff(fact_6516_boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,uminus_uminus(A),X)) = top_top(A) ) ).

% boolean_algebra.disj_cancel_right
tff(fact_6517_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y2)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_6518_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y2)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_6519_Compl__Diff__eq,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),B5) ).

% Compl_Diff_eq
tff(fact_6520_Inf__sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B5: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B5)),A3) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B5))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),A3) = top_top(A) ) ) ) ) ).

% Inf_sup_eq_top_iff
tff(fact_6521_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A,Z: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)))) ) ).

% distrib_sup_le
tff(fact_6522_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A,Z: A] : pp(aa(A,bool,aa(A,fun(A,bool),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),X),Y2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z)))) ) ).

% distrib_inf_le
tff(fact_6523_sup__inf__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y2: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z),X)) ) ).

% sup_inf_distrib2
tff(fact_6524_sup__inf__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)) ) ).

% sup_inf_distrib1
tff(fact_6525_inf__sup__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y2: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z),X)) ) ).

% inf_sup_distrib2
tff(fact_6526_inf__sup__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z)) ) ).

% inf_sup_distrib1
tff(fact_6527_distrib__imp2,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A,Z: A] :
          ( ! [X2: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y3),Z3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Z3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z)) ) ) ) ).

% distrib_imp2
tff(fact_6528_distrib__imp1,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A,Z: A] :
          ( ! [X2: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y3),Z3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Z3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)) ) ) ) ).

% distrib_imp1
tff(fact_6529_Un__Diff__Int,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5)) = A4 ).

% Un_Diff_Int
tff(fact_6530_Int__Diff__Un,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = A4 ).

% Int_Diff_Un
tff(fact_6531_Diff__Int,axiom,
    ! [A: $tType,A4: set(A),B5: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),C5)) ).

% Diff_Int
tff(fact_6532_Diff__Un,axiom,
    ! [A: $tType,A4: set(A),B5: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),C5)) ).

% Diff_Un
tff(fact_6533_Compl__Un,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).

% Compl_Un
tff(fact_6534_Compl__Int,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).

% Compl_Int
tff(fact_6535_Compl__partition2,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),A4) = top_top(set(A)) ).

% Compl_partition2
tff(fact_6536_Compl__partition,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = top_top(set(A)) ).

% Compl_partition
tff(fact_6537_sup__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),B2)) = top_top(A) ) ).

% sup_cancel_left2
tff(fact_6538_sup__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),B2)) = top_top(A) ) ).

% sup_cancel_left1
tff(fact_6539_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_abv(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P))),collect(A,Q)) ).

% Collect_imp_eq
tff(fact_6540_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) ) ).

% inf_sup_aci(8)
tff(fact_6541_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)) ) ).

% inf_sup_aci(7)
tff(fact_6542_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z)) ) ).

% inf_sup_aci(6)
tff(fact_6543_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),X) ) ).

% inf_sup_aci(5)
tff(fact_6544_sup_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).

% sup.assoc
tff(fact_6545_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z)) ) ).

% sup_assoc
tff(fact_6546_sup_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: 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),B2),A3) ) ).

% sup.commute
tff(fact_6547_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),X) ) ).

% sup_commute
tff(fact_6548_sup_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).

% sup.left_commute
tff(fact_6549_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y2: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)) ) ).

% sup_left_commute
tff(fact_6550_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F2: fun(A,B),G: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F2),G),X3) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) ) ).

% sup_fun_def
tff(fact_6551_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% less_supI1
tff(fact_6552_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% less_supI2
tff(fact_6553_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb3
tff(fact_6554_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_6555_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% sup.strict_boundedE
tff(fact_6556_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6557_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.strict_coboundedI1
tff(fact_6558_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.strict_coboundedI2
tff(fact_6559_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_6560_Un__Diff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C5)) ).

% Un_Diff
tff(fact_6561_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y2: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2))) ) ).

% inf_sup_ord(4)
tff(fact_6562_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2))) ) ).

% inf_sup_ord(3)
tff(fact_6563_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X)) ) ) ) ).

% le_supE
tff(fact_6564_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X)) ) ) ) ).

% le_supI
tff(fact_6565_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2))) ) ).

% sup_ge1
tff(fact_6566_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y2: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2))) ) ).

% sup_ge2
tff(fact_6567_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% le_supI1
tff(fact_6568_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% le_supI2
tff(fact_6569_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,D2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_6570_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_6571_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y2: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z)),X)) ) ) ) ).

% sup_least
tff(fact_6572_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) = Y2 ) ) ) ).

% le_iff_sup
tff(fact_6573_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.orderE
tff(fact_6574_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) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% sup.orderI
tff(fact_6575_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y2: A] :
          ( ! [X2: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F2,X2),Y3)))
         => ( ! [X2: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F2,X2),Y3)))
           => ( ! [X2: A,Y3: A,Z3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X2))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),X2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y3),Z3)),X2)) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) = aa(A,A,aa(A,fun(A,A),F2,X),Y2) ) ) ) ) ) ).

% sup_unique
tff(fact_6576_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb1
tff(fact_6577_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_6578_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) = X ) ) ) ).

% sup_absorb1
tff(fact_6579_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) = Y2 ) ) ) ).

% sup_absorb2
tff(fact_6580_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).

% sup.boundedE
tff(fact_6581_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)) ) ) ) ).

% sup.boundedI
tff(fact_6582_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6583_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).

% sup.cobounded1
tff(fact_6584_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).

% sup.cobounded2
tff(fact_6585_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6586_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6587_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.coboundedI1
tff(fact_6588_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.coboundedI2
tff(fact_6589_Diff__partition,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)) = B5 ) ) ).

% Diff_partition
tff(fact_6590_Diff__subset__conv,axiom,
    ! [A: $tType,A4: set(A),B5: set(A),C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),C5))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C5))) ) ).

% Diff_subset_conv
tff(fact_6591_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F2: fun(A,B),A4: A,B5: A] :
          ( order_mono(A,B,F2)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A4)),aa(A,B,F2,B5))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B5)))) ) ) ).

% mono_sup
tff(fact_6592_sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B5: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),B5)) = aa(set(A),A,complete_Inf_Inf(A),image(A,A,aa(A,fun(A,A),sup_sup(A),A3),B5)) ) ).

% sup_Inf
tff(fact_6593_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),B5: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,B5))),A3) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_abw(fun(B,A),fun(A,fun(B,A)),F2),A3),B5)) ) ).

% INF_sup
tff(fact_6594_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B5: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B5)),A3) = aa(set(A),A,complete_Inf_Inf(A),image(A,A,aTP_Lamp_abx(A,fun(A,A),A3),B5)) ) ).

% Inf_sup
tff(fact_6595_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F2: fun(B,A),B5: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,B5))) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aby(A,fun(fun(B,A),fun(B,A)),A3),F2),B5)) ) ).

% sup_INF
tff(fact_6596_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),A4: set(B),G: fun(C,A),B5: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))),aa(set(A),A,complete_Inf_Inf(A),image(C,A,G,B5))) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_aca(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B5),A4)) ) ).

% INF_sup_distrib2
tff(fact_6597_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) = top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y2)) ) ) ).

% sup_shunt
tff(fact_6598_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y2)),Z))) ) ) ).

% shunt1
tff(fact_6599_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y2))),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z))) ) ) ).

% shunt2
tff(fact_6600_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P2: A,Q5: A,R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q5),R)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P2),aa(A,A,uminus_uminus(A),Q5))),R)) ) ) ).

% sup_neg_inf
tff(fact_6601_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),X) = Y2 ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_6602_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_6603_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
     => ? [H2: fun(A,A)] : bij_betw(A,A,H2,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_6604_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_6605_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_6606_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_6607_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6608_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_6609_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: A,B5: 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_aax(A,fun(nat,A),B5),collect(nat,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B5) ) ).

% SUP_nat_binary
tff(fact_6610_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_acb(A,fun(A,bool)),aTP_Lamp_acc(A,fun(A,bool))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_6611_sum_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),B5: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,finite_finite(B),B5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5)))
                 => ( aa(B,A,G,X2) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B5)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_6612_sum__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A4: set(B),B5: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,finite_finite(B),B5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5))) ) ) ) ) ).

% sum_Un
tff(fact_6613_prod_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),B5: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,finite_finite(B),B5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5)))
                 => ( aa(B,A,G,X2) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = 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),G),B5)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6614_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => boolea2506097494486148201lgebra(A,inf_inf(A),sup_sup(A),uminus_uminus(A),bot_bot(A),top_top(A)) ) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_6615_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B5: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)))
         => ( aa(set(A),B,aa(fun(A,B),fun(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),B5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(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),B5))),aa(set(A),B,aa(fun(A,B),fun(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)),B5),A4)))),aa(set(A),B,aa(fun(A,B),fun(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),B5))) ) ) ) ).

% sum_Un2
tff(fact_6616_sum_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B),B5: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,finite_finite(B),B5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A4)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5))) ) ) ) ) ).

% sum.union_diff2
tff(fact_6617_prod_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B),B5: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,finite_finite(B),B5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),times_times(A),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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A4)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5))) ) ) ) ) ).

% prod.union_diff2
tff(fact_6618_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B5: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5))
    <=> ( inj_on(A,B,F2,A4)
        & inj_on(A,B,F2,B5)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),image(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5))),image(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_6619_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_6620_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6621_sum__Un__nat,axiom,
    ! [A: $tType,A4: set(A),B5: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( pp(aa(set(A),bool,finite_finite(A),B5))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(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),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B5))),aa(set(A),nat,aa(fun(A,nat),fun(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),B5))) ) ) ) ).

% sum_Un_nat
tff(fact_6622_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_6623_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6624_prod__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [A4: set(B),B5: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( pp(aa(set(B),bool,finite_finite(B),B5))
           => ( ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5)))
                 => ( aa(B,A,F2,X2) != zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),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),F2),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B5))) ) ) ) ) ) ).

% prod_Un
tff(fact_6625_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),I: B,B5: set(A),J4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),fun_upd(B,set(A),A4,I,B5),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),A4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J4),aa(set(B),set(B),insert(B,I),bot_bot(set(B))))))),if(set(A),aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),J4),B5,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_6626_times__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,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))),bool),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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lp(nat,fun(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_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% times_int.rsp
tff(fact_6627_intrel__iff,axiom,
    ! [X: nat,Y2: nat,U: nat,V2: nat] :
      ( pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Y2)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,U),V2)))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y2) ) ) ).

% intrel_iff
tff(fact_6628_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_6629_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_6630_zero__int_Orsp,axiom,
    pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat)))) ).

% zero_int.rsp
tff(fact_6631_int_Oabs__eq__iff,axiom,
    ! [X: product_prod(nat,nat),Y2: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),int,abs_Integ,X) = aa(product_prod(nat,nat),int,abs_Integ,Y2) )
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,X),Y2)) ) ).

% int.abs_eq_iff
tff(fact_6632_uminus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_lq(nat,fun(nat,product_prod(nat,nat))))),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_lq(nat,fun(nat,product_prod(nat,nat)))))) ).

% uminus_int.rsp
tff(fact_6633_nat_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(product_prod(nat,nat),nat),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),nat,nat,intrel,fequal(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))),product_case_prod(nat,nat,nat,minus_minus(nat)))) ).

% nat.rsp
tff(fact_6634_one__int_Orsp,axiom,
    pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat)))) ).

% one_int.rsp
tff(fact_6635_of__int_Orsp,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(fun(product_prod(nat,nat),A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(product_prod(nat,nat),A),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),A,A,intrel,fequal(A)),product_case_prod(nat,nat,A,aTP_Lamp_lr(nat,fun(nat,A)))),product_case_prod(nat,nat,A,aTP_Lamp_lr(nat,fun(nat,A))))) ) ).

% of_int.rsp
tff(fact_6636_intrel__def,axiom,
    intrel = product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ace(nat,fun(nat,fun(product_prod(nat,nat),bool)))) ).

% intrel_def
tff(fact_6637_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,X: B,A4: set(B),F2: fun(B,A),Y2: A] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A4))
       => ( image(B,A,fun_upd(B,A,F2,X,Y2),A4) = aa(set(A),set(A),insert(A,Y2),image(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) )
      & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A4))
       => ( image(B,A,fun_upd(B,A,F2,X,Y2),A4) = image(B,A,F2,A4) ) ) ) ).

% fun_upd_image
tff(fact_6638_less__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lt(nat,fun(nat,fun(product_prod(nat,nat),bool))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lt(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_int.rsp
tff(fact_6639_less__eq__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lv(nat,fun(nat,fun(product_prod(nat,nat),bool))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lv(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_eq_int.rsp
tff(fact_6640_int_Orel__eq__transfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),intrel),fequal(int))) ).

% int.rel_eq_transfer
tff(fact_6641_minus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,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))),bool),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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lz(nat,fun(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_lz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% minus_int.rsp
tff(fact_6642_plus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,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))),bool),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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lx(nat,fun(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_lx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% plus_int.rsp
tff(fact_6643_inverse__real_Oabs__eq,axiom,
    ! [X: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X),X))
     => ( aa(real,real,inverse_inverse(real),aa(fun(nat,rat),real,real2,X)) = aa(fun(nat,rat),real,real2,if(fun(nat,rat),vanishes(X),aTP_Lamp_abq(nat,rat),aTP_Lamp_abr(fun(nat,rat),fun(nat,rat),X))) ) ) ).

% inverse_real.abs_eq
tff(fact_6644_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_6645_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,X: A,X3: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_acf(A,option(B)),X,none(B)),X3) = none(B) ).

% empty_upd_none
tff(fact_6646_nth__Cons__Suc,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).

% nth_Cons_Suc
tff(fact_6647_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_6648_take__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : take(A,aa(nat,nat,suc,N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,N,Xs)) ).

% take_Suc_Cons
tff(fact_6649_enumerate__simps_I2_J,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : enumerate(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(product_prod(nat,B)),list(product_prod(nat,B)),aa(product_prod(nat,B),fun(list(product_prod(nat,B)),list(product_prod(nat,B))),cons(product_prod(nat,B)),aa(B,product_prod(nat,B),product_Pair(nat,B,N),X)),enumerate(B,aa(nat,nat,suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_6650_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M2,A3) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),M2,A3,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),insert(A,B2),ran(B,A,M2)) ) ) ).

% ran_map_upd
tff(fact_6651_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xs: list(A),V2: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_6652_take__Cons__numeral,axiom,
    ! [A: $tType,V2: num,X: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)),Xs)) ).

% take_Cons_numeral
tff(fact_6653_Cons__in__lex,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y2: A,Ys2: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(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),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))),lex(A,R)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),R))
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) ) )
        | ( ( X = Y2 )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2)),lex(A,R))) ) ) ) ).

% Cons_in_lex
tff(fact_6654_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,T2: fun(A,option(B)),K: A,X: B] :
      ~ ! [X2: A] : aa(A,option(B),fun_upd(A,option(B),T2,K,aa(B,option(B),some(B),X)),X2) = none(B) ).

% map_upd_nonempty
tff(fact_6655_real_Oabs__induct,axiom,
    ! [P: fun(real,bool),X: real] :
      ( ! [Y3: fun(nat,rat)] :
          ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,Y3),Y3))
         => pp(aa(real,bool,P,aa(fun(nat,rat),real,real2,Y3))) )
     => pp(aa(real,bool,P,X)) ) ).

% real.abs_induct
tff(fact_6656_replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : replicate(A,aa(nat,nat,suc,N),X) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),replicate(A,N,X)) ).

% replicate_Suc
tff(fact_6657_list__update__code_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I: nat,Y2: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(nat,nat,suc,I),Y2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),list_update(A,Xs,I,Y2)) ).

% list_update_code(3)
tff(fact_6658_list__update__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y2: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),zero_zero(nat),Y2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs) ).

% list_update_code(2)
tff(fact_6659_of__rat__Real,axiom,
    ! [X: rat] : aa(rat,real,field_char_0_of_rat(real),X) = aa(fun(nat,rat),real,real2,aTP_Lamp_os(rat,fun(nat,rat),X)) ).

% of_rat_Real
tff(fact_6660_length__Cons,axiom,
    ! [A: $tType,X: 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),X),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_6661_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y4: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).

% length_Suc_conv
tff(fact_6662_Suc__length__conv,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,N) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y4: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).

% Suc_length_conv
tff(fact_6663_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2)))
     => ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2) ) ) ).

% impossible_Cons
tff(fact_6664_zero__real__def,axiom,
    zero_zero(real) = aa(fun(nat,rat),real,real2,aTP_Lamp_abq(nat,rat)) ).

% zero_real_def
tff(fact_6665_one__real__def,axiom,
    one_one(real) = aa(fun(nat,rat),real,real2,aTP_Lamp_abt(nat,rat)) ).

% one_real_def
tff(fact_6666_of__int__Real,axiom,
    ! [X: int] : aa(int,real,ring_1_of_int(real),X) = aa(fun(nat,rat),real,real2,aTP_Lamp_acg(int,fun(nat,rat),X)) ).

% of_int_Real
tff(fact_6667_Suc__le__length__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ? [X4: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys3))) ) ) ).

% Suc_le_length_iff
tff(fact_6668_of__nat__Real,axiom,
    ! [X: nat] : aa(nat,real,semiring_1_of_nat(real),X) = aa(fun(nat,rat),real,real2,aTP_Lamp_ach(nat,fun(nat,rat),X)) ).

% of_nat_Real
tff(fact_6669_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y2: A,Xs: list(A)] :
      ( ( ( X = Y2 )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y2)),one_one(nat)) ) )
      & ( ( X != Y2 )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y2) = aa(A,nat,count_list(A,Xs),Y2) ) ) ) ).

% count_list.simps(2)
tff(fact_6670_uminus__real_Oabs__eq,axiom,
    ! [X: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X),X))
     => ( aa(real,real,uminus_uminus(real),aa(fun(nat,rat),real,real2,X)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ov(fun(nat,rat),fun(nat,rat)),X)) ) ) ).

% uminus_real.abs_eq
tff(fact_6671_plus__real_Oabs__eq,axiom,
    ! [Xa: fun(nat,rat),X: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,Xa),Xa))
     => ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X),X))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(fun(nat,rat),real,real2,Xa)),aa(fun(nat,rat),real,real2,X)) = 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_ow(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Xa),X)) ) ) ) ).

% plus_real.abs_eq
tff(fact_6672_times__real_Oabs__eq,axiom,
    ! [Xa: fun(nat,rat),X: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,Xa),Xa))
     => ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X),X))
       => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(fun(nat,rat),real,real2,Xa)),aa(fun(nat,rat),real,real2,X)) = 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_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Xa),X)) ) ) ) ).

% times_real.abs_eq
tff(fact_6673_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X22: 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),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_6674_nth__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% nth_Cons'
tff(fact_6675_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X22: list(A)] : size_list(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),size_list(A,X,X22))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6676_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I)),J2))
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J2)) = 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),J2))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_6677_map__of__zip__upd,axiom,
    ! [A: $tType,B: $tType,Ys2: list(B),Xs: list(A),Zs: list(B),X: A,Y2: B,Z: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Ys2) = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(A),nat,size_size(list(A)),Xs) )
       => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => ( ( fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),X,aa(B,option(B),some(B),Y2)) = fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Zs)),X,aa(B,option(B),some(B),Z)) )
           => ( map_of(A,B,zip(A,B,Xs,Ys2)) = map_of(A,B,zip(A,B,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_6678_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),J2))
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J2)) = 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),J2))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_6679_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6680_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y2: A,Xs: list(A),N: nat] :
      ( ( X != Y2 )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = Y2 )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y2 )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6681_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat,Y2: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = replicate(A,N,Y2) )
    <=> ( ( X = Y2 )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_6682_Cons__lenlex__iff,axiom,
    ! [A: $tType,M2: A,Ms: list(A),N: A,Ns: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(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),M2),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns))),lenlex(A,R)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,M2),N)),R)) )
        | ( ( M2 = N )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ms),Ns)),lenlex(A,R))) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6683_Real_Opositive_Oabs__eq,axiom,
    ! [X: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X),X))
     => ( pp(aa(real,bool,positive2,aa(fun(nat,rat),real,real2,X)))
      <=> ? [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
            & ? [K3: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,X,N5))) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_6684_upto__aux__rec,axiom,
    ! [J2: int,I: int,Js: list(int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I))
       => ( upto_aux(I,J2,Js) = Js ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I))
       => ( upto_aux(I,J2,Js) = upto_aux(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J2),Js)) ) ) ) ).

% upto_aux_rec
tff(fact_6685_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_real,real2),aTP_Lamp_abs(fun(nat,rat),fun(nat,rat))) ).

% inverse_real_def
tff(fact_6686_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_real,real2),aTP_Lamp_ov(fun(nat,rat),fun(nat,rat))) ).

% uminus_real_def
tff(fact_6687_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,M2: fun(B,option(A)),X: B,Y2: A,Z: A] :
      ( ( aa(B,option(A),M2,X) = aa(A,option(A),some(A),Y2) )
     => ( inj_on(B,option(A),M2,dom(B,A,M2))
       => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),ran(B,A,M2)))
         => ( ran(B,A,fun_upd(B,option(A),M2,X,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,M2)),aa(set(A),set(A),insert(A,Y2),bot_bot(set(A))))),aa(set(A),set(A),insert(A,Z),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_6688_le__Real,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y5)))
        <=> ! [R5: rat] :
              ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
             => ? [K3: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(nat,rat,X6,N5)),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Y5,N5)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_6689_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( ( dom(A,B,F2) = bot_bot(set(A)) )
    <=> ! [X4: A] : aa(A,option(B),F2,X4) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_6690_fun__upd__None__if__notin__dom,axiom,
    ! [B: $tType,A: $tType,K: A,M2: fun(A,option(B))] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),K),dom(A,B,M2)))
     => ( fun_upd(A,option(B),M2,K,none(B)) = M2 ) ) ).

% fun_upd_None_if_notin_dom
tff(fact_6691_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_acf(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_6692_dom__map__of__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( dom(A,B,map_of(A,B,zip(A,B,Xs,Ys2))) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% dom_map_of_zip
tff(fact_6693_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y2: option(B),F2: fun(A,option(B)),X: A] :
      ( ( ( Y2 = none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F2,X,Y2)) = 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,X),bot_bot(set(A)))) ) )
      & ( ( Y2 != none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F2,X,Y2)) = aa(set(A),set(A),insert(A,X),dom(A,B,F2)) ) ) ) ).

% dom_fun_upd
tff(fact_6694_realrel__refl,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X6),X6)) ) ).

% realrel_refl
tff(fact_6695_cauchy__add,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ow(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y5)) ) ) ).

% cauchy_add
tff(fact_6696_cauchy__const,axiom,
    ! [X: rat] : cauchy(aTP_Lamp_os(rat,fun(nat,rat),X)) ).

% cauchy_const
tff(fact_6697_cauchy__minus,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => cauchy(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ov(fun(nat,rat),fun(nat,rat)),X6)) ) ).

% cauchy_minus
tff(fact_6698_cauchy__mult,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y5)) ) ) ).

% cauchy_mult
tff(fact_6699_cauchy__diff,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ox(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y5)) ) ) ).

% cauchy_diff
tff(fact_6700_domIff,axiom,
    ! [A: $tType,B: $tType,A3: A,M2: fun(A,option(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),dom(A,B,M2)))
    <=> ( aa(A,option(B),M2,A3) != none(B) ) ) ).

% domIff
tff(fact_6701_dom__def,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : dom(A,B,M2) = collect(A,aTP_Lamp_aci(fun(A,option(B)),fun(A,bool),M2)) ).

% dom_def
tff(fact_6702_Real__induct,axiom,
    ! [P: fun(real,bool),X: real] :
      ( ! [X8: fun(nat,rat)] :
          ( cauchy(X8)
         => pp(aa(real,bool,P,aa(fun(nat,rat),real,real2,X8))) )
     => pp(aa(real,bool,P,X)) ) ).

% Real_induct
tff(fact_6703_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( pp(aa(set(A),bool,finite_finite(A),dom(A,B,F2)))
     => ( ~ pp(aa(set(A),bool,finite_finite(A),top_top(set(A))))
       => ? [X2: A] : aa(A,option(B),F2,X2) = none(B) ) ) ).

% finite_map_freshness
tff(fact_6704_cr__real__eq,axiom,
    ! [X3: fun(nat,rat),Xa2: real] :
      ( pp(aa(real,bool,aa(fun(nat,rat),fun(real,bool),pcr_real,X3),Xa2))
    <=> ( cauchy(X3)
        & ( aa(fun(nat,rat),real,real2,X3) = Xa2 ) ) ) ).

% cr_real_eq
tff(fact_6705_dom__minus,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),X: B,A4: set(B)] :
      ( ( aa(B,option(A),F2,X) = 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,X),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_6706_cauchy__inverse,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ pp(vanishes(X6))
       => cauchy(aTP_Lamp_abr(fun(nat,rat),fun(nat,rat),X6)) ) ) ).

% cauchy_inverse
tff(fact_6707_cauchy__imp__bounded,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ? [B4: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B4))
          & ! [N4: nat] : pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N4))),B4)) ) ) ).

% cauchy_imp_bounded
tff(fact_6708_less__RealD,axiom,
    ! [Y5: fun(nat,rat),X: real] :
      ( cauchy(Y5)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(fun(nat,rat),real,real2,Y5)))
       => ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(rat,real,field_char_0_of_rat(real),aa(nat,rat,Y5,N2)))) ) ) ).

% less_RealD
tff(fact_6709_le__RealI,axiom,
    ! [Y5: fun(nat,rat),X: real] :
      ( cauchy(Y5)
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(rat,real,field_char_0_of_rat(real),aa(nat,rat,Y5,N2))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(fun(nat,rat),real,real2,Y5))) ) ) ).

% le_RealI
tff(fact_6710_Real__leI,axiom,
    ! [X6: fun(nat,rat),Y2: real] :
      ( cauchy(X6)
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(rat,real,field_char_0_of_rat(real),aa(nat,rat,X6,N2))),Y2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(fun(nat,rat),real,real2,X6)),Y2)) ) ) ).

% Real_leI
tff(fact_6711_minus__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( 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_ov(fun(nat,rat),fun(nat,rat)),X6)) ) ) ).

% minus_Real
tff(fact_6712_add__Real,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y5)) = 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_ow(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y5)) ) ) ) ).

% add_Real
tff(fact_6713_mult__Real,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y5)) = 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_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y5)) ) ) ) ).

% mult_Real
tff(fact_6714_diff__Real,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y5)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ox(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y5)) ) ) ) ).

% diff_Real
tff(fact_6715_realrelI,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ox(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y5)))
         => pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X6),Y5)) ) ) ) ).

% realrelI
tff(fact_6716_eq__Real,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( ( aa(fun(nat,rat),real,real2,X6) = aa(fun(nat,rat),real,real2,Y5) )
        <=> pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ox(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y5))) ) ) ) ).

% eq_Real
tff(fact_6717_vanishes__diff__inverse,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ pp(vanishes(X6))
       => ( cauchy(Y5)
         => ( ~ pp(vanishes(Y5))
           => ( pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ox(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y5)))
             => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_acj(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y5))) ) ) ) ) ) ).

% vanishes_diff_inverse
tff(fact_6718_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),P: fun(fun(A,option(B)),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),dom(A,B,M2)))
     => ( pp(aa(fun(A,option(B)),bool,P,aTP_Lamp_acf(A,option(B))))
       => ( ! [K2: A,V4: B,M3: fun(A,option(B))] :
              ( pp(aa(set(A),bool,finite_finite(A),dom(A,B,M3)))
             => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),K2),dom(A,B,M3)))
               => ( pp(aa(fun(A,option(B)),bool,P,M3))
                 => pp(aa(fun(A,option(B)),bool,P,fun_upd(A,option(B),M3,K2,aa(B,option(B),some(B),V4)))) ) ) )
         => pp(aa(fun(A,option(B)),bool,P,M2)) ) ) ) ).

% finite_Map_induct
tff(fact_6719_realrel__def,axiom,
    ! [X3: fun(nat,rat),Xa2: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X3),Xa2))
    <=> ( cauchy(X3)
        & cauchy(Xa2)
        & pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ox(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X3),Xa2))) ) ) ).

% realrel_def
tff(fact_6720_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),X: A] :
      ( ( dom(A,B,F2) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
    <=> ? [V5: B] : F2 = fun_upd(A,option(B),aTP_Lamp_acf(A,option(B)),X,aa(B,option(B),some(B),V5)) ) ).

% dom_eq_singleton_conv
tff(fact_6721_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ pp(vanishes(X6))
       => ? [B4: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B4))
            & ? [K2: nat] :
                ( ! [N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N4))
                   => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B4),aa(rat,rat,uminus_uminus(rat),aa(nat,rat,X6,N4)))) )
                | ! [N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N4))
                   => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B4),aa(nat,rat,X6,N4))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_6722_positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( pp(aa(real,bool,positive2,aa(fun(nat,rat),real,real2,X6)))
      <=> ? [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
            & ? [K3: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,X6,N5))) ) ) ) ) ).

% positive_Real
tff(fact_6723_cauchy__not__vanishes,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ pp(vanishes(X6))
       => ? [B4: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B4))
            & ? [K2: nat] :
              ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N4))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B4),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N4)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_6724_cauchy__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
    <=> ! [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
         => ? [K3: nat] :
            ! [M5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),M5))
             => ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),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,N5)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_6725_cauchyI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3))
         => ? [K4: nat] :
            ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K4),M3))
             => ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K4),N2))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M3)),aa(nat,rat,X6,N2)))),R3)) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_6726_cauchyD,axiom,
    ! [X6: fun(nat,rat),R: rat] :
      ( cauchy(X6)
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R))
       => ? [K2: nat] :
          ! [M: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M))
           => ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N4))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),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,N4)))),R)) ) ) ) ) ).

% cauchyD
tff(fact_6727_inverse__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ( pp(vanishes(X6))
         => ( aa(real,real,inverse_inverse(real),aa(fun(nat,rat),real,real2,X6)) = zero_zero(real) ) )
        & ( ~ pp(vanishes(X6))
         => ( aa(real,real,inverse_inverse(real),aa(fun(nat,rat),real,real2,X6)) = aa(fun(nat,rat),real,real2,aTP_Lamp_abr(fun(nat,rat),fun(nat,rat),X6)) ) ) ) ) ).

% inverse_Real
tff(fact_6728_not__positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ pp(aa(real,bool,positive2,aa(fun(nat,rat),real,real2,X6)))
      <=> ! [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
           => ? [K3: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(nat,rat,X6,N5)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_6729_dom__override__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),G: fun(A,option(B)),A4: set(A)] : dom(A,B,override_on(A,option(B),F2,G,A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F2)),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_ack(fun(A,option(B)),fun(set(A),fun(A,bool)),G),A4)))),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_acl(fun(A,option(B)),fun(set(A),fun(A,bool)),G),A4))) ).

% dom_override_on
tff(fact_6730_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),Xs: list(A),Ys2: list(B)] : dom(A,B,map_upds(A,B,M2,Xs,Ys2)) = 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)),Ys2),Xs))),dom(A,B,M2)) ).

% dom_map_upds
tff(fact_6731_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,M2: fun(A,option(B)),Ys2: list(B),Y2: B] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I))
     => ( map_upds(A,B,M2,Xs,list_update(B,Ys2,I,Y2)) = map_upds(A,B,M2,Xs,Ys2) ) ) ).

% map_upds_list_update2_drop
tff(fact_6732_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,X: A,Ys2: list(B),Xs: list(A),F2: fun(A,option(B)),Y2: B] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F2,X,aa(B,option(B),some(B),Y2)),Xs,Ys2) = map_upds(A,B,F2,Xs,Ys2) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F2,X,aa(B,option(B),some(B),Y2)),Xs,Ys2) = fun_upd(A,option(B),map_upds(A,B,F2,Xs,Ys2),X,aa(B,option(B),some(B),Y2)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6733_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),X: A,Y2: B] : restrict_map(A,B,fun_upd(A,option(B),M2,X,aa(B,option(B),some(B),Y2)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = restrict_map(A,B,M2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_6734_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),D6: set(A),M2: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D6))
       => ( restrict_map(A,B,map_upds(A,B,M2,Xs,Ys2),D6) = map_upds(A,B,restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(list(A),set(A),set2(A),Xs))),Xs,Ys2) ) ) ) ).

% restrict_map_upds
tff(fact_6735_restrict__out,axiom,
    ! [A: $tType,B: $tType,X: A,A4: set(A),M2: fun(A,option(B))] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
     => ( aa(A,option(B),restrict_map(A,B,M2,A4),X) = none(B) ) ) ).

% restrict_out
tff(fact_6736_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D6: set(A),X3: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_acf(A,option(B)),D6),X3) = none(B) ).

% restrict_map_empty
tff(fact_6737_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),X3: A] : aa(A,option(B),restrict_map(A,B,M2,bot_bot(set(A))),X3) = none(B) ).

% restrict_map_to_empty
tff(fact_6738_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,X: A,D6: set(A),M2: fun(A,option(B)),Y2: option(B)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
       => ( restrict_map(A,B,fun_upd(A,option(B),M2,X,Y2),D6) = fun_upd(A,option(B),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),X,Y2) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
       => ( restrict_map(A,B,fun_upd(A,option(B),M2,X,Y2),D6) = restrict_map(A,B,M2,D6) ) ) ) ).

% restrict_fun_upd
tff(fact_6739_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,X: A,D6: set(A),M2: fun(A,option(B)),Y2: option(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
     => ( fun_upd(A,option(B),restrict_map(A,B,M2,D6),X,Y2) = fun_upd(A,option(B),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),X,Y2) ) ) ).

% fun_upd_restrict_conv
tff(fact_6740_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,X: A,D6: set(A),M2: fun(A,option(B))] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
       => ( fun_upd(A,option(B),restrict_map(A,B,M2,D6),X,none(B)) = restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
       => ( fun_upd(A,option(B),restrict_map(A,B,M2,D6),X,none(B)) = restrict_map(A,B,M2,D6) ) ) ) ).

% fun_upd_None_restrict
tff(fact_6741_restrict__map__def,axiom,
    ! [A: $tType,B: $tType,A4: set(A),M2: fun(A,option(B)),X3: A] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A4))
       => ( aa(A,option(B),restrict_map(A,B,M2,A4),X3) = aa(A,option(B),M2,X3) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A4))
       => ( aa(A,option(B),restrict_map(A,B,M2,A4),X3) = none(B) ) ) ) ).

% restrict_map_def
tff(fact_6742_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),D6: set(A),X: A,Y2: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M2,D6),X,Y2) = fun_upd(A,option(B),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),X,Y2) ).

% fun_upd_restrict
tff(fact_6743_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),X: A] : restrict_map(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = fun_upd(A,option(B),F2,X,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_6744_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_real,map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real,real2)),aTP_Lamp_oq(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% times_real_def
tff(fact_6745_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_real,map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real,real2)),aTP_Lamp_ow(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% plus_real_def
tff(fact_6746_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list(nat)] : aa(list(nat),nat,nat_list_encode,aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),aa(list(nat),nat,nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_6747_cr__real__def,axiom,
    ! [X3: fun(nat,rat),Xa2: real] :
      ( pp(aa(real,bool,aa(fun(nat,rat),fun(real,bool),cr_real,X3),Xa2))
    <=> ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X3),X3))
        & ( aa(fun(nat,rat),real,real2,X3) = Xa2 ) ) ) ).

% cr_real_def
tff(fact_6748_list__encode__eq,axiom,
    ! [X: list(nat),Y2: list(nat)] :
      ( ( aa(list(nat),nat,nat_list_encode,X) = aa(list(nat),nat,nat_list_encode,Y2) )
    <=> ( X = Y2 ) ) ).

% list_encode_eq
tff(fact_6749_bij__list__encode,axiom,
    bij_betw(list(nat),nat,nat_list_encode,top_top(set(list(nat))),top_top(set(nat))) ).

% bij_list_encode
tff(fact_6750_inj__list__encode,axiom,
    ! [A4: set(list(nat))] : inj_on(list(nat),nat,nat_list_encode,A4) ).

% inj_list_encode
tff(fact_6751_surj__list__encode,axiom,
    image(list(nat),nat,nat_list_encode,top_top(set(list(nat)))) = top_top(set(nat)) ).

% surj_list_encode
tff(fact_6752_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
tff(fact_6753_list__encode_Oelims,axiom,
    ! [X: list(nat),Y2: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,X) = Y2 )
     => ( ( ( X = nil(nat) )
         => ( Y2 != zero_zero(nat) ) )
       => ~ ! [X2: nat,Xs2: list(nat)] :
              ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs2) )
             => ( Y2 != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X2),aa(list(nat),nat,nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_6754_lex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_acm(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% lex_conv
tff(fact_6755_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys2: 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),X),nil(A)),Ys2))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% concat_inth
tff(fact_6756_append__eq__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys2: 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)),Ys2) )
        | ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
     => ( ( append(A,Xs,Us) = append(A,Ys2,Vs) )
      <=> ( ( Xs = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_6757_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_6758_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(list(A),nat,size_size(list(A)),append(A,Xs,Ys2)) = 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)),Ys2)) ).

% length_append
tff(fact_6759_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( aa(list(A),A,groups8242544230860333062m_list(A),nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_6760_take0,axiom,
    ! [A: $tType,X3: list(A)] : take(A,zero_zero(nat),X3) = nil(A) ).

% take0
tff(fact_6761_take__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( take(A,N,Xs) = nil(A) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_6762_take__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,N,Xs) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_6763_empty__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( nil(A) = replicate(A,N,X) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_6764_replicate__empty,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( replicate(A,N,X) = nil(A) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_6765_zip__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Us: list(B),Ys2: 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,Ys2),append(B,Us,Vs)) = append(product_prod(A,B),zip(A,B,Xs,Us),zip(A,B,Ys2,Vs)) ) ) ).

% zip_append
tff(fact_6766_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: 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),nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_6767_fun__upds__append2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),M2: fun(A,option(B)),Zs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( map_upds(A,B,M2,Xs,append(B,Ys2,Zs)) = map_upds(A,B,M2,Xs,Ys2) ) ) ).

% fun_upds_append2_drop
tff(fact_6768_fun__upds__append__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),M2: fun(A,option(B)),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( map_upds(A,B,M2,append(A,Xs,Zs),Ys2) = map_upds(A,B,M2,Xs,Ys2) ) ) ).

% fun_upds_append_drop
tff(fact_6769_n__lists__Nil,axiom,
    ! [A: $tType,N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) )
      & ( ( N != zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).

% n_lists_Nil
tff(fact_6770_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_6771_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys2: list(A)] : aa(nat,A,nth(A,append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% nth_append_length
tff(fact_6772_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),N: nat] : aa(nat,A,nth(A,append(A,Xs,Ys2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys2),N) ).

% nth_append_length_plus
tff(fact_6773_take__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys2: list(A)] : take(A,N,append(A,Xs,Ys2)) = append(A,take(A,N,Xs),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys2)) ).

% take_append
tff(fact_6774_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys2: list(A),Y2: A] : list_update(A,append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2)),aa(list(A),nat,size_size(list(A)),Xs),Y2) = append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)) ).

% list_update_length
tff(fact_6775_nths__singleton,axiom,
    ! [A: $tType,A4: set(nat),X: A] :
      ( ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A4) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) )
      & ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A4) = nil(A) ) ) ) ).

% nths_singleton
tff(fact_6776_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,set_ord_lessThan(nat,aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,set_ord_lessThan(nat,K)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_6777_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,set_ord_atMost(nat,aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,set_ord_atMost(nat,K)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_6778_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),M2: fun(A,option(B)),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2)))
     => ( map_upds(A,B,M2,append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),Ys2) = fun_upd(A,option(B),map_upds(A,B,M2,Xs,Ys2),X,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_6779_list__encode_Osimps_I1_J,axiom,
    aa(list(nat),nat,nat_list_encode,nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_6780_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : size_list(A,X,nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_6781_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_6782_replicate__0,axiom,
    ! [A: $tType,X: A] : replicate(A,zero_zero(nat),X) = nil(A) ).

% replicate_0
tff(fact_6783_rotate__append,axiom,
    ! [A: $tType,L: list(A),Q5: list(A)] : aa(list(A),list(A),rotate(A,aa(list(A),nat,size_size(list(A)),L)),append(A,L,Q5)) = append(A,Q5,L) ).

% rotate_append
tff(fact_6784_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(A) )
       => ( ( append(A,Xs,Ys2) = append(A,Ys2,Xs) )
         => ? [N2: nat,Zs2: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N2))
              & ( concat(A,replicate(list(A),N2,Zs2)) = append(A,Xs,Ys2) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_6785_enumerate__append__eq,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys2: list(A)] : enumerate(A,N,append(A,Xs,Ys2)) = append(product_prod(nat,A),enumerate(A,N,Xs),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys2)) ).

% enumerate_append_eq
tff(fact_6786_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_6787_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,bool)] : find(A,Uu,nil(A)) = none(A) ).

% find.simps(1)
tff(fact_6788_nths__Cons,axiom,
    ! [A: $tType,X: A,L: list(A),A4: set(nat)] : nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L),A4) = append(A,if(list(A),aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),nil(A)),nths(A,L,collect(nat,aTP_Lamp_acn(set(nat),fun(nat,bool),A4)))) ).

% nths_Cons
tff(fact_6789_list__encode_Ocases,axiom,
    ! [X: list(nat)] :
      ( ( X != nil(nat) )
     => ~ ! [X2: nat,Xs2: list(nat)] : X != aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs2) ) ).

% list_encode.cases
tff(fact_6790_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_6791_same__length__different,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs != Ys2 )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
       => ? [Pre: list(A),X2: A,Xs4: list(A),Y3: A,Ys4: list(A)] :
            ( ( X2 != Y3 )
            & ( Xs = append(A,Pre,append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)),Xs4)) )
            & ( Ys2 = append(A,Pre,append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A)),Ys4)) ) ) ) ) ).

% same_length_different
tff(fact_6792_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y4: A,Ys3: list(A)] :
          ( ( Xs = append(A,Ys3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_6793_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys2: list(B),Zs: list(C),Ws2: list(D),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D),bool))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys2) = 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) )
         => ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,nil(A)),nil(B)),nil(C)),nil(D)))
           => ( ! [X2: A,Xs2: list(A),Y3: B,Ys5: 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)),Ys5) )
                 => ( ( aa(list(B),nat,size_size(list(B)),Ys5) = 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) )
                     => ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,Xs2),Ys5),Zs2),Ws))
                       => pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5)),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))) ) ) ) )
             => pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,Xs),Ys2),Zs),Ws2)) ) ) ) ) ) ).

% list_induct4
tff(fact_6794_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys2: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),bool)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys2) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,nil(A)),nil(B)),nil(C)))
         => ( ! [X2: A,Xs2: list(A),Y3: B,Ys5: list(B),Z3: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys5) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys5) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,Xs2),Ys5),Zs2))
                   => pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2))) ) ) )
           => pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,Xs),Ys2),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_6795_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),P: fun(list(A),fun(list(B),bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
       => ( ! [X2: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys5) )
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys5))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5))) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys2)) ) ) ) ).

% list_induct2
tff(fact_6796_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_6797_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y2: A] : aa(A,nat,count_list(A,nil(A)),Y2) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_6798_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6799_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Ys2: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( list_update(A,append(A,Xs,Ys2),I,X) = append(A,list_update(A,Xs,I,X),Ys2) ) ) ).

% list_update_append1
tff(fact_6800_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys2: list(A),Qs: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),append(A,Xs,Zs)),append(A,Ys2,Qs))),lexord(A,R)))
     => ( ( Xs != Ys2 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
         => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
           => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2)),lexord(A,R))) ) ) ) ) ).

% lexord_sufE
tff(fact_6801_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2)),lex(A,R)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),append(A,Xs,Us)),append(A,Ys2,Vs))),lex(A,R))) ) ) ).

% lex_append_rightI
tff(fact_6802_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R2: set(product_prod(A,A)),Vs: list(A),Ys2: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(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)),Ys2) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),append(A,Us,Vs)),append(A,Xs,Ys2))),lenlex(A,R2))) ) ) ).

% lenlex_append1
tff(fact_6803_nths__append,axiom,
    ! [A: $tType,L: list(A),L6: list(A),A4: set(nat)] : nths(A,append(A,L,L6),A4) = append(A,nths(A,L,A4),nths(A,L6,collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_aco(list(A),fun(set(nat),fun(nat,bool)),L),A4)))) ).

% nths_append
tff(fact_6804_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_acp(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ).

% lexordp.mono
tff(fact_6805_nth__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys2: list(A)] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,append(A,Xs,Ys2)),N) = aa(nat,A,nth(A,Xs),N) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,append(A,Xs,Ys2)),N) = aa(nat,A,nth(A,Ys2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ) ).

% nth_append
tff(fact_6806_list__update__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys2: list(A),X: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,append(A,Xs,Ys2),N,X) = append(A,list_update(A,Xs,N,X),Ys2) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,append(A,Xs,Ys2),N,X) = append(A,Xs,list_update(A,Ys2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),X)) ) ) ) ).

% list_update_append
tff(fact_6807_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W: list(A),R: set(product_prod(A,A)),V2: list(A),Z: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),U),W)),lexord(A,R)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W)),aa(list(A),nat,size_size(list(A)),U)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),append(A,U,V2)),append(A,W,Z))),lexord(A,R))) ) ) ).

% lexord_sufI
tff(fact_6808_remdups__adj__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( remdups_adj(A,replicate(A,N,X)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( remdups_adj(A,replicate(A,N,X)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ) ).

% remdups_adj_replicate
tff(fact_6809_remdups__adj__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( remdups_adj(A,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
     => ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),X) ) ) ).

% remdups_adj_singleton
tff(fact_6810_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: A,Xs: list(B),Ys2: 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,Ys2)) = 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),Ys2))) ) ).

% horner_sum_append
tff(fact_6811_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6812_take__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs)) ) ) ) ).

% take_Cons'
tff(fact_6813_nth__repl,axiom,
    ! [A: $tType,M2: nat,Xs: list(A),N: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( M2 != N )
         => ( aa(nat,A,nth(A,append(A,take(A,N,Xs),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),Xs)))),M2) = aa(nat,A,nth(A,Xs),M2) ) ) ) ) ).

% nth_repl
tff(fact_6814_pos__n__replace,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Y2: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),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,N,Xs),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A)),drop(A,aa(nat,nat,suc,N),Xs)))) ) ) ).

% pos_n_replace
tff(fact_6815_drop0,axiom,
    ! [A: $tType,X3: list(A)] : drop(A,zero_zero(nat),X3) = X3 ).

% drop0
tff(fact_6816_map__of__eq__empty__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ! [X4: A] : aa(A,option(B),map_of(A,B,Xys),X4) = none(B)
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% map_of_eq_empty_iff
tff(fact_6817_empty__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ( aTP_Lamp_acf(A,option(B)) = map_of(A,B,Xys) )
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% empty_eq_map_of_iff
tff(fact_6818_drop__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : drop(A,aa(nat,nat,suc,N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,N,Xs) ).

% drop_Suc_Cons
tff(fact_6819_length__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),drop(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_drop
tff(fact_6820_drop__update__cancel,axiom,
    ! [A: $tType,N: nat,M2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( drop(A,M2,list_update(A,Xs,N,X)) = drop(A,M2,Xs) ) ) ).

% drop_update_cancel
tff(fact_6821_drop__replicate,axiom,
    ! [A: $tType,I: nat,K: nat,X: A] : drop(A,I,replicate(A,K,X)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I),X) ).

% drop_replicate
tff(fact_6822_drop__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,N,Xs) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_6823_drop__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( drop(A,N,Xs) = nil(A) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_6824_drop__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( drop(A,N,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_6825_drop__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys2: list(A)] : drop(A,N,append(A,Xs,Ys2)) = append(A,drop(A,N,Xs),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys2)) ).

% drop_append
tff(fact_6826_drop__Cons__numeral,axiom,
    ! [A: $tType,V2: num,X: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)),Xs) ).

% drop_Cons_numeral
tff(fact_6827_nth__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,drop(A,N,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I)) ) ) ).

% nth_drop
tff(fact_6828_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_6829_map__of_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType,X3: A] : aa(A,option(B),map_of(A,B,nil(product_prod(A,B))),X3) = none(B) ).

% map_of.simps(1)
tff(fact_6830_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_6831_drop__take,axiom,
    ! [A: $tType,N: nat,M2: nat,Xs: list(A)] : drop(A,N,take(A,M2,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),drop(A,N,Xs)) ).

% drop_take
tff(fact_6832_append__eq__conv__conj,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( append(A,Xs,Ys2) = Zs )
    <=> ( ( Xs = take(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) )
        & ( Ys2 = drop(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) ) ) ) ).

% append_eq_conv_conj
tff(fact_6833_drop__update__swap,axiom,
    ! [A: $tType,M2: nat,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( drop(A,M2,list_update(A,Xs,N,X)) = list_update(A,drop(A,M2,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2),X) ) ) ).

% drop_update_swap
tff(fact_6834_drop__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
      & ( ( N != zero_zero(nat) )
       => ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ) ).

% drop_Cons'
tff(fact_6835_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) )
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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) ) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( 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_6836_zip__append2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),Zs: list(B)] : zip(A,B,Xs,append(B,Ys2,Zs)) = append(product_prod(A,B),zip(A,B,take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs),Ys2),zip(A,B,drop(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs),Zs)) ).

% zip_append2
tff(fact_6837_zip__append1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(A),Zs: list(B)] : zip(A,B,append(A,Xs,Ys2),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,Ys2,drop(B,aa(list(A),nat,size_size(list(A)),Xs),Zs))) ).

% zip_append1
tff(fact_6838_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6839_rotate__drop__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,N),Xs) = append(A,drop(A,modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)),Xs),take(A,modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)),Xs)) ).

% rotate_drop_take
tff(fact_6840_id__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6841_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A),A3: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6842_lexn__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] : lexn(A,R,N) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_acq(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R),N))) ).

% lexn_conv
tff(fact_6843_upto_Opsimps,axiom,
    ! [I: int,J2: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,I),J2))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J2))
         => ( upto(I,J2) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J2)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J2))
         => ( upto(I,J2) = nil(int) ) ) ) ) ).

% upto.psimps
tff(fact_6844_upto__Nil,axiom,
    ! [I: int,J2: int] :
      ( ( upto(I,J2) = nil(int) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I)) ) ).

% upto_Nil
tff(fact_6845_upto__Nil2,axiom,
    ! [I: int,J2: int] :
      ( ( nil(int) = upto(I,J2) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I)) ) ).

% upto_Nil2
tff(fact_6846_upto__empty,axiom,
    ! [J2: int,I: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I))
     => ( upto(I,J2) = nil(int) ) ) ).

% upto_empty
tff(fact_6847_nth__upto,axiom,
    ! [I: int,K: nat,J2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J2))
     => ( aa(nat,int,nth(int,upto(I,J2)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_6848_length__upto,axiom,
    ! [I: int,J2: int] : aa(list(int),nat,size_size(list(int)),upto(I,J2)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),I)),one_one(int))) ).

% length_upto
tff(fact_6849_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M2),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M2)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M2)),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M2),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(1)
tff(fact_6850_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = 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),M2))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(4)
tff(fact_6851_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) = 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),M2))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(3)
tff(fact_6852_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M2)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M2)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(2)
tff(fact_6853_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lexn(A,R,zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_6854_upto__split2,axiom,
    ! [I: int,J2: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J2),K))
       => ( upto(I,K) = append(int,upto(I,J2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J2),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_6855_upto__split1,axiom,
    ! [I: int,J2: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J2),K))
       => ( upto(I,K) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int))),upto(J2,K)) ) ) ) ).

% upto_split1
tff(fact_6856_atLeastLessThan__upto,axiom,
    ! [I: int,J2: int] : set_or7035219750837199246ssThan(int,I,J2) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6857_greaterThanAtMost__upto,axiom,
    ! [I: int,J2: int] : set_or3652927894154168847AtMost(int,I,J2) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J2)) ).

% greaterThanAtMost_upto
tff(fact_6858_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2)),lexn(A,R,N)))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
        & ( aa(list(A),nat,size_size(list(A)),Ys2) = N ) ) ) ).

% lexn_length
tff(fact_6859_upto__rec1,axiom,
    ! [I: int,J2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J2))
     => ( upto(I,J2) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J2)) ) ) ).

% upto_rec1
tff(fact_6860_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y2: list(int)] :
      ( ( upto(X,Xa) = Y2 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
         => ( Y2 = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
         => ( Y2 = nil(int) ) ) ) ) ).

% upto.elims
tff(fact_6861_upto_Osimps,axiom,
    ! [I: int,J2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J2))
       => ( upto(I,J2) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J2)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J2))
       => ( upto(I,J2) = nil(int) ) ) ) ).

% upto.simps
tff(fact_6862_upto__rec2,axiom,
    ! [I: int,J2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J2))
     => ( upto(I,J2) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J2),nil(int))) ) ) ).

% upto_rec2
tff(fact_6863_greaterThanLessThan__upto,axiom,
    ! [I: int,J2: int] : set_or5935395276787703475ssThan(int,I,J2) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_6864_upto__split3,axiom,
    ! [I: int,J2: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J2),K))
       => ( upto(I,K) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J2),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_6865_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y2: list(int)] :
      ( ( upto(X,Xa) = Y2 )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,X),Xa))
       => ~ ( ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
               => ( Y2 = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
               => ( Y2 = nil(int) ) ) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,X),Xa)) ) ) ) ).

% upto.pelims
tff(fact_6866_take__hd__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( append(A,take(A,N,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,drop(A,N,Xs))),nil(A))) = take(A,aa(nat,nat,suc,N),Xs) ) ) ).

% take_hd_drop
tff(fact_6867_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),K: A,V2: B] : graph(A,B,fun_upd(A,option(B),M2,K,aa(B,option(B),some(B),V2))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,K),V2)),graph(A,B,fun_upd(A,option(B),M2,K,none(B)))) ).

% graph_map_upd
tff(fact_6868_hd__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( hd(A,replicate(A,N,X)) = X ) ) ).

% hd_replicate
tff(fact_6869_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_acf(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_6870_hd__take,axiom,
    ! [A: $tType,J2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),J2))
     => ( hd(A,take(A,J2,Xs)) = hd(A,Xs) ) ) ).

% hd_take
tff(fact_6871_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_6872_hd__drop__conv__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( hd(A,drop(A,N,Xs)) = aa(nat,A,nth(A,Xs),N) ) ) ).

% hd_drop_conv_nth
tff(fact_6873_graph__fun__upd__None,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),K: A] : graph(A,B,fun_upd(A,option(B),M2,K,none(B))) = collect(product_prod(A,B),aa(A,fun(product_prod(A,B),bool),aTP_Lamp_acr(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),M2),K)) ).

% graph_fun_upd_None
tff(fact_6874_hd__rotate__conv__nth,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( hd(A,aa(list(A),list(A),rotate(A,N),Xs)) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% hd_rotate_conv_nth
tff(fact_6875_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_6876_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat)] :
      ( ( ( Xs = nil(A) )
       => ( size_list(A,F2,Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( size_list(A,F2,Xs) = 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_6877_list__encode_Opelims,axiom,
    ! [X: list(nat),Y2: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,X) = Y2 )
     => ( accp(list(nat),nat_list_encode_rel,X)
       => ( ( ( X = nil(nat) )
           => ( ( Y2 = zero_zero(nat) )
             => ~ accp(list(nat),nat_list_encode_rel,nil(nat)) ) )
         => ~ ! [X2: nat,Xs2: list(nat)] :
                ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs2) )
               => ( ( Y2 = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X2),aa(list(nat),nat,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),X2),Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_6878_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_6879_tl__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : tl(A,replicate(A,N,X)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ).

% tl_replicate
tff(fact_6880_drop__Suc,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,aa(nat,nat,suc,N),Xs) = drop(A,N,tl(A,Xs)) ).

% drop_Suc
tff(fact_6881_take__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : take(A,N,tl(A,Xs)) = tl(A,take(A,aa(nat,nat,suc,N),Xs)) ).

% take_tl
tff(fact_6882_tl__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : tl(A,take(A,N,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),tl(A,Xs)) ).

% tl_take
tff(fact_6883_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),tl(A,Xs))) ) ) ) ).

% Nitpick.size_list_simp(2)
tff(fact_6884_nth__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),tl(A,Xs))))
     => ( aa(nat,A,nth(A,tl(A,Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).

% nth_tl
tff(fact_6885_take__Suc,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( take(A,aa(nat,nat,suc,N),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,Xs)),take(A,N,tl(A,Xs))) ) ) ).

% take_Suc
tff(fact_6886_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(num,nat,numeral_numeral(nat),M2)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N)) = nil(nat) ) ) ) ).

% upt_rec_numeral
tff(fact_6887_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( A4 != bot_bot(set(A)) )
           => ( linord4507533701916653071of_set(A,A4) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),lattic643756798350308766er_Min(A,A4)),linord4507533701916653071of_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,lattic643756798350308766er_Min(A,A4)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_6888_tl__upt,axiom,
    ! [M2: nat,N: nat] : tl(nat,upt(M2,N)) = upt(aa(nat,nat,suc,M2),N) ).

% tl_upt
tff(fact_6889_hd__upt,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ( hd(nat,upt(I,J2)) = I ) ) ).

% hd_upt
tff(fact_6890_length__upt,axiom,
    ! [I: nat,J2: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I,J2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I) ).

% length_upt
tff(fact_6891_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( A4 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),lattic643756798350308766er_Min(A,A4)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_6892_upt__eq__Nil__conv,axiom,
    ! [I: nat,J2: nat] :
      ( ( upt(I,J2) = nil(nat) )
    <=> ( ( J2 = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_6893_nth__upt,axiom,
    ! [I: nat,K: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J2))
     => ( aa(nat,nat,nth(nat,upt(I,J2)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).

% nth_upt
tff(fact_6894_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_6895_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list(nat),Q5: nat] :
      ( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),M2),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns)) = upt(M2,Q5) )
    <=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns) = upt(aa(nat,nat,suc,M2),Q5) ) ) ).

% upt_conv_Cons_Cons
tff(fact_6896_greaterThanAtMost__upt,axiom,
    ! [N: nat,M2: nat] : set_or3652927894154168847AtMost(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),aa(nat,nat,suc,M2))) ).

% greaterThanAtMost_upt
tff(fact_6897_atLeast__upt,axiom,
    ! [N: nat] : set_ord_lessThan(nat,N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),N)) ).

% atLeast_upt
tff(fact_6898_atLeastAtMost__upt,axiom,
    ! [N: nat,M2: nat] : set_or1337092689740270186AtMost(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(N,aa(nat,nat,suc,M2))) ).

% atLeastAtMost_upt
tff(fact_6899_greaterThanLessThan__upt,axiom,
    ! [N: nat,M2: nat] : set_or5935395276787703475ssThan(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),M2)) ).

% greaterThanLessThan_upt
tff(fact_6900_atMost__upto,axiom,
    ! [N: nat] : set_ord_atMost(nat,N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,N))) ).

% atMost_upto
tff(fact_6901_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( A4 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),lattic643756798350308766er_Min(A,A4)),X))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).

% Min_less_iff
tff(fact_6902_upt__conv__Cons,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ( upt(I,J2) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J2)) ) ) ).

% upt_conv_Cons
tff(fact_6903_enumerate__eq__zip,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,N,Xs) = zip(nat,A,upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_6904_upt__eq__Cons__conv,axiom,
    ! [I: nat,J2: nat,X: nat,Xs: list(nat)] :
      ( ( upt(I,J2) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
        & ( I = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J2) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_6905_upt__rec,axiom,
    ! [I: nat,J2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
       => ( upt(I,J2) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
       => ( upt(I,J2) = nil(nat) ) ) ) ).

% upt_rec
tff(fact_6906_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
           => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( lattic643756798350308766er_Min(A,A4) = X ) )
              & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( lattic643756798350308766er_Min(A,A4) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(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,X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Min.remove
tff(fact_6907_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),insert(A,X),A4)) = X ) )
            & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),insert(A,X),A4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(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,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Min.insert_remove
tff(fact_6908_upt__Suc,axiom,
    ! [I: nat,J2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
       => ( upt(I,aa(nat,nat,suc,J2)) = append(nat,upt(I,J2),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J2),nil(nat))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
       => ( upt(I,aa(nat,nat,suc,J2)) = nil(nat) ) ) ) ).

% upt_Suc
tff(fact_6909_upt__Suc__append,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
     => ( upt(I,aa(nat,nat,suc,J2)) = append(nat,upt(I,J2),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J2),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_6910_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] : aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),map(nat,bool,bit_se5641148757651400278ts_bit(A,A3),upt(zero_zero(nat),N))) = bit_se2584673776208193580ke_bit(A,N,A3) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_6911_map__fst__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : map(product_prod(nat,A),nat,product_fst(nat,A),enumerate(A,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_6912_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_6913_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aTP_Lamp_acs(B,A),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_6914_nth__map,axiom,
    ! [B: $tType,A: $tType,N: nat,Xs: list(A),F2: fun(A,B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,B,nth(B,map(A,B,F2,Xs)),N) = aa(A,B,F2,aa(nat,A,nth(A,Xs),N)) ) ) ).

% nth_map
tff(fact_6915_map__fst__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( map(product_prod(A,B),A,product_fst(A,B),zip(A,B,Xs,Ys2)) = Xs ) ) ).

% map_fst_zip
tff(fact_6916_map__snd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( map(product_prod(A,B),B,product_snd(A,B),zip(A,B,Xs,Ys2)) = Ys2 ) ) ).

% map_snd_zip
tff(fact_6917_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(B,A),Xs: list(B),G: fun(C,A),Ys2: list(C)] :
      ( ( map(B,A,F2,Xs) = map(C,A,G,Ys2) )
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys2) ) ) ).

% map_eq_imp_length_eq
tff(fact_6918_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : map(B,A,aTP_Lamp_act(A,fun(B,A),K),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_6919_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_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_cf(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),Xs)) = aa(A,A,aa(A,fun(A,A),minus_minus(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_subtractf
tff(fact_6920_uminus__sum__list__map,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),Xs: list(B)] : aa(A,A,uminus_uminus(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F2,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),comp(A,A,B,uminus_uminus(A)),F2),Xs)) ) ).

% uminus_sum_list_map
tff(fact_6921_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I: nat] : map(nat,A,aTP_Lamp_acu(A,fun(nat,A),X),upt(zero_zero(nat),I)) = replicate(A,I,X) ).

% map_replicate_trivial
tff(fact_6922_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) )
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X2)),aa(A,B,G,X2))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),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_6923_zip__eq__conv,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: 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)),Ys2) )
     => ( ( zip(A,B,Xs,Ys2) = Zs )
      <=> ( ( map(product_prod(A,B),A,product_fst(A,B),Zs) = Xs )
          & ( map(product_prod(A,B),B,product_snd(A,B),Zs) = Ys2 ) ) ) ) ).

% zip_eq_conv
tff(fact_6924_sum__list__triv,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [R: B,Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),map(C,B,aTP_Lamp_acv(B,fun(C,B),R),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(list(C),nat,size_size(list(C)),Xs))),R) ) ).

% sum_list_triv
tff(fact_6925_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),N: nat] : map(nat,A,F2,upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F2,zero_zero(nat))),map(nat,A,aTP_Lamp_wi(fun(nat,A),fun(nat,A),F2),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_6926_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_6927_size__list__conv__sum__list,axiom,
    ! [B: $tType,F2: fun(B,nat),Xs: list(B)] : size_list(B,F2,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(B,nat,F2,Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% size_list_conv_sum_list
tff(fact_6928_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_lg(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_6929_nth__map__upt,axiom,
    ! [A: $tType,I: nat,N: nat,M2: nat,F2: fun(nat,A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))
     => ( aa(nat,A,nth(A,map(nat,A,F2,upt(M2,N))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),I)) ) ) ).

% nth_map_upt
tff(fact_6930_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F2: fun(A,B),X3: A] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,map(A,B,F2,Xs))),X3) = aa(B,option(B),some(B),aa(A,B,F2,X3)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,map(A,B,F2,Xs))),X3) = none(B) ) ) ) ).

% map_of_zip_map
tff(fact_6931_map__fst__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : map(product_prod(A,B),A,product_fst(A,B),zip(A,B,Xs,Ys2)) = 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)),Ys2)),Xs) ).

% map_fst_zip_take
tff(fact_6932_map__snd__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(B),Ys2: list(A)] : map(product_prod(B,A),A,product_snd(B,A),zip(B,A,Xs,Ys2)) = 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)),Ys2)),Ys2) ).

% map_snd_zip_take
tff(fact_6933_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,M2: 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),N),M2) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),I3)) ) )
       => ( map(nat,A,F2,upt(M2,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_6934_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),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),I3)) = N ) )
       => ( transpose(A,Xs) = map(nat,list(A),aTP_Lamp_acx(list(list(A)),fun(nat,list(A)),Xs),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_6935_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [F2: fun(nat,B),Ns: list(nat)] :
          ( ! [X2: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(nat,B,F2,X2)),aa(nat,B,F2,Y3))) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(B),B,groups8242544230860333062m_list(B),map(nat,B,F2,Ns)))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_6936_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] : map(nat,nat,suc,upt(M2,N)) = upt(aa(nat,nat,suc,M2),aa(nat,nat,suc,N)) ).

% map_Suc_upt
tff(fact_6937_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys2)
           => ( ( aa(list(A),set(A),set2(A),Ys2) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys2 = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_6938_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2))
        <=> ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ys2)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) )
            & sorted_wrt(A,ord_less(A),Ys2) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_6939_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = concat(list(A),map(list(A),list(list(A)),aTP_Lamp_acz(list(A),fun(list(A),list(list(A))),Xs),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_6940_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6941_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I2: nat,J3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_6942_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),I: nat,J2: nat] :
      ( sorted_wrt(A,P,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J2))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_6943_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_6944_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_6945_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_6946_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_6947_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : sorted_wrt(nat,ord_less(nat),upt(M2,N)) ).

% sorted_wrt_upt
tff(fact_6948_length__concat,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(B),nat,size_size(list(B)),concat(B,Xss)) = aa(list(nat),nat,groups8242544230860333062m_list(nat),map(list(B),nat,size_size(list(B)),Xss)) ).

% length_concat
tff(fact_6949_map__add__upt,axiom,
    ! [N: nat,M2: nat] : map(nat,nat,aTP_Lamp_ada(nat,fun(nat,nat),N),upt(zero_zero(nat),M2)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) ).

% map_add_upt
tff(fact_6950_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6951_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I2: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_6952_map__decr__upt,axiom,
    ! [M2: nat,N: nat] : map(nat,nat,aTP_Lamp_kz(nat,nat),upt(aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = upt(M2,N) ).

% map_decr_upt
tff(fact_6953_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I2)))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_6954_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ~ ! [L2: list(A)] :
                ( sorted_wrt(A,ord_less(A),L2)
               => ( ( aa(list(A),set(A),set2(A),L2) = A4 )
                 => ( aa(list(A),nat,size_size(list(A)),L2) != aa(set(A),nat,finite_card(A),A4) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_6955_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J2: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J2))) ) ) ) ) ).

% sorted_nth_mono
tff(fact_6956_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I2: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_6957_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_6958_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),L: list(A)] :
          ( pp(aa(set(A),bool,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_6959_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S3: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S3))
       => ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ~ ! [L2: list(B)] :
                ( sorted_wrt(A,Less,map(B,A,F2,L2))
               => ( ( aa(list(B),set(B),set2(B),L2) = A4 )
                 => ( aa(list(B),nat,size_size(list(B)),L2) != aa(set(B),nat,finite_card(B),A4) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_6960_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)))
     => ( ( ( Xs = nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = zero_zero(nat) ) )
        & ( ( Xs != nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat))) ) ) ) ) ).

% length_transpose_sorted
tff(fact_6961_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_6962_sorted__wrt__upto,axiom,
    ! [I: int,J2: int] : sorted_wrt(int,ord_less(int),upto(I,J2)) ).

% sorted_wrt_upto
tff(fact_6963_zip__rev,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( zip(A,B,rev(A,Xs),rev(B,Ys2)) = rev(product_prod(A,B),zip(A,B,Xs,Ys2)) ) ) ).

% zip_rev
tff(fact_6964_take__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : take(A,N,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)),N),Xs)) ).

% take_rev
tff(fact_6965_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_6966_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_6967_drop__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,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)),N),Xs)) ).

% drop_rev
tff(fact_6968_rotate__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,N),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,N,aa(list(A),nat,size_size(list(A)),Xs)))),Xs)) ).

% rotate_rev
tff(fact_6969_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_6970_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_me(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_6971_rev__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,rev(A,Xs)),N) = 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,N))) ) ) ).

% rev_nth
tff(fact_6972_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y2: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( rev(A,list_update(A,Xs,K,Y2)) = 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)),Y2) ) ) ).

% rev_update
tff(fact_6973_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I2))),aa(nat,A,nth(A,Xs),I2))) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_6974_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I2: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I2))) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_6975_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J2: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J2)),aa(nat,A,nth(A,Xs),I))) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_6976_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_adb(nat,fun(list(A),bool),I),Xs))))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J2) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J2)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_6977_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)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( map(list(A),A,aTP_Lamp_adc(nat,fun(list(A),A),I),filter2(list(A),aTP_Lamp_adb(nat,fun(list(A),bool),I),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_6978_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_6979_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),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,bool),comp(A,bool,B,P),F2),Xs)) ).

% length_filter_map
tff(fact_6980_length__filter__less,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P,X))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6981_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),filter2(A,aa(A,fun(A,bool),fequal(A),X),Xs)),X) = filter2(A,aa(A,fun(A,bool),fequal(A),X),Xs) ).

% replicate_length_filter
tff(fact_6982_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_filter_le
tff(fact_6983_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),filter2(A,aTP_Lamp_ab(fun(A,bool),fun(A,bool),P),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_6984_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F2: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F2,filter2(B,P,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_add(fun(B,A),fun(fun(B,bool),fun(B,A)),F2),P),Xs)) ) ).

% sum_list_map_filter'
tff(fact_6985_sum__list__map__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B),P: fun(B,bool),F2: fun(B,A)] :
          ( ! [X2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),aa(list(B),set(B),set2(B),Xs)))
             => ( ~ pp(aa(B,bool,P,X2))
               => ( aa(B,A,F2,X2) = zero_zero(A) ) ) )
         => ( aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F2,filter2(B,P,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F2,Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_6986_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,Y2),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_ade(A,fun(A,bool),Y2),Xs)) ).

% set_minus_filter_out
tff(fact_6987_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : filter2(A,P,Xs) = nths(A,Xs,collect(nat,aa(list(A),fun(nat,bool),aTP_Lamp_adf(fun(A,bool),fun(list(A),fun(nat,bool)),P),Xs))) ).

% filter_eq_nths
tff(fact_6988_length__filter__conv__card,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs)) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(list(A),fun(nat,bool),aTP_Lamp_adf(fun(A,bool),fun(list(A),fun(nat,bool)),P2),Xs))) ).

% length_filter_conv_card
tff(fact_6989_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( 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)),collect(A,P)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_6990_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_adc(nat,fun(list(A),A),I),filter2(list(A),aTP_Lamp_adb(nat,fun(list(A),bool),I),Xs)) ) ) ).

% nth_transpose
tff(fact_6991_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)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_adb(nat,fun(list(A),bool),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_6992_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S3: set(B),F2: fun(B,A),A4: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S3))
       => ( pp(aa(set(B),bool,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_6993_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_adg(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_adh(list(A),bool),Xs)) ).

% transpose_max_length
tff(fact_6994_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_6995_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P: fun(nat,bool),Xs: list(A),Is: list(nat)] : map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_adi(fun(nat,bool),fun(product_prod(A,nat),bool),P),zip(A,nat,Xs,Is))) = map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_adj(fun(nat,bool),fun(product_prod(A,nat),bool),P),zip(A,nat,Xs,map(nat,nat,suc,Is)))) ).

% nths_shift_lemma_Suc
tff(fact_6996_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_adk(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A3),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_6997_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S3: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),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_6998_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_adl(set(nat),fun(product_prod(A,nat),bool),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),bool),aTP_Lamp_adm(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A4),I),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_6999_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_adl(set(nat),fun(product_prod(A,nat),bool),A4),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_7000_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_adg(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_7001_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y2: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( ( ( Xs = nil(A) )
             => ( foldr(A,A,ord_max(A),Xs,Y2) = Y2 ) )
            & ( ( Xs != nil(A) )
             => ( foldr(A,A,ord_max(A),Xs,Y2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y2) ) ) ) ) ) ).

% foldr_max_sorted
tff(fact_7002_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_adn(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_ado(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_adp(list(B),bool),Xss),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_7003_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S3: set(B),F2: fun(B,A),X: B,A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),insert(B,X),A4)),S3))
       => ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))) = remove1(B,X,sorted8670434370408473282of_set(A,B,Less_eq,F2,A4)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_7004_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S3: set(B),F2: fun(B,A),X: B,A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),insert(B,X),A4)),S3))
       => ( pp(aa(set(B),bool,finite_finite(B),A4))
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),insert(B,X),A4)) = insort_key(A,B,Less_eq,F2,X,sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_7005_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,bool),Xs: list(B)] : map(B,A,F2,filter2(B,P,Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_adq(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F2),P),Xs) ).

% map_filter_map_filter
tff(fact_7006_length__product__lists,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),product_lists(B,Xss)) = foldr(nat,nat,times_times(nat),map(list(B),nat,size_size(list(B)),Xss),one_one(nat)) ).

% length_product_lists
tff(fact_7007_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_adh(list(A),bool),Xs) ) ) ).

% transpose_transpose
tff(fact_7008_map__filter__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F2,Xs) = map(A,B,aa(fun(A,option(B)),fun(A,B),comp(option(B),B,A,the2(B)),F2),filter2(A,aTP_Lamp_aci(fun(A,option(B)),fun(A,bool),F2),Xs)) ).

% map_filter_def
tff(fact_7009_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_7010_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_7011_takeWhile__eq__take,axiom,
    ! [A: $tType,P: fun(A,bool),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_7012_length__takeWhile__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_7013_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_7014_option_Osel,axiom,
    ! [A: $tType,X23: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X23)) = X23 ).

% option.sel
tff(fact_7015_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_7016_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_7017_option_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,Option: option(A),F1: B,F22: fun(A,B)] :
      ( ( ( Option = none(A) )
       => ( aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option) = F1 ) )
      & ( ( Option != none(A) )
       => ( aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option) = aa(A,B,F22,aa(option(A),A,the2(A),Option)) ) ) ) ).

% option.case_eq_if
tff(fact_7018_takeWhile__nth,axiom,
    ! [A: $tType,J2: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))))
     => ( aa(nat,A,nth(A,takeWhile(A,P,Xs)),J2) = aa(nat,A,nth(A,Xs),J2) ) ) ).

% takeWhile_nth
tff(fact_7019_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)))
     => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))))) ) ).

% nth_length_takeWhile
tff(fact_7020_Option_Othese__def,axiom,
    ! [A: $tType,A4: set(option(A))] : these(A,A4) = image(option(A),A,the2(A),collect(option(A),aTP_Lamp_ni(set(option(A)),fun(option(A),bool),A4))) ).

% Option.these_def
tff(fact_7021_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
         => ( lattic643756798350308766er_Min(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Min.infinite
tff(fact_7022_option_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)))
    <=> ~ ( ( ( Option = none(A) )
            & ~ pp(aa(B,bool,P,F1)) )
          | ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
            & ~ pp(aa(B,bool,P,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel_asm
tff(fact_7023_option_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)))
    <=> ( ( ( Option = none(A) )
         => pp(aa(B,bool,P,F1)) )
        & ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
         => pp(aa(B,bool,P,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel
tff(fact_7024_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J2: nat,P: fun(A,bool),Xs: list(A)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_7025_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) ) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
         => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) )
       => ( takeWhile(A,P,Xs) = take(A,N,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_7026_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),T2: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,map(B,A,F2,Xs)))
         => ( filter2(B,aa(A,fun(B,bool),aTP_Lamp_adr(fun(B,A),fun(A,fun(B,bool)),F2),T2),Xs) = takeWhile(B,aa(A,fun(B,bool),aTP_Lamp_adr(fun(B,A),fun(A,fun(B,bool)),F2),T2),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_7027_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : lattic643756798350308766er_Min(A,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_ads(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Min.eq_fold'
tff(fact_7028_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( ( S3 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),lattic643756798349783984er_Max(A,S3)) = lattic643756798350308766er_Min(A,image(A,A,uminus_uminus(A),S3)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_7029_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( lattic643756798349783984er_Max(nat,collect(nat,aTP_Lamp_jk(nat,fun(nat,bool),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_7030_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero(int) )
     => ( lattic643756798349783984er_Max(int,collect(int,aTP_Lamp_ka(int,fun(int,bool),N))) = aa(int,int,abs_abs(int),N) ) ) ).

% Max_divisors_self_int
tff(fact_7031_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( A4 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),lattic643756798349783984er_Max(A,A4)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).

% Max_less_iff
tff(fact_7032_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( ( S3 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),lattic643756798350308766er_Min(A,S3)) = lattic643756798349783984er_Max(A,image(A,A,uminus_uminus(A),S3)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_7033_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : lattic643756798349783984er_Max(A,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_adt(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Max.eq_fold'
tff(fact_7034_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( A4 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),lattic643756798349783984er_Max(A,A4)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_7035_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
         => ( lattic643756798349783984er_Max(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_7036_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M2: int] :
      ( ( N != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M2),N) = lattic643756798349783984er_Max(int,collect(int,aa(int,fun(int,bool),aTP_Lamp_adu(int,fun(int,fun(int,bool)),N),M2))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_7037_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(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_7038_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_7039_card__le__Suc__Max,axiom,
    ! [S3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S3)),aa(nat,nat,suc,lattic643756798349783984er_Max(nat,S3)))) ) ).

% card_le_Suc_Max
tff(fact_7040_Sup__nat__def,axiom,
    ! [X6: set(nat)] :
      ( ( ( X6 = bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = zero_zero(nat) ) )
      & ( ( X6 != bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = lattic643756798349783984er_Max(nat,X6) ) ) ) ).

% Sup_nat_def
tff(fact_7041_divide__nat__def,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M2),N) = lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_adv(nat,fun(nat,fun(nat,bool)),N),M2))) ) ) ) ).

% divide_nat_def
tff(fact_7042_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N) = lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_adw(nat,fun(nat,fun(nat,bool)),N),M2))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_7043_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
           => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( lattic643756798349783984er_Max(A,A4) = X ) )
              & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( lattic643756798349783984er_Max(A,A4) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(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,X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Max.remove
tff(fact_7044_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),insert(A,X),A4)) = X ) )
            & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),insert(A,X),A4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(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,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Max.insert_remove
tff(fact_7045_Gcd__eq__Max,axiom,
    ! [M10: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),M10))
     => ( ( M10 != bot_bot(set(nat)) )
       => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M10))
         => ( gcd_Gcd(nat,M10) = lattic643756798349783984er_Max(nat,aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),image(nat,set(nat),aTP_Lamp_adx(nat,set(nat)),M10))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_7046_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_ady(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Sup_fin.eq_fold'
tff(fact_7047_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_adz(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Inf_fin.eq_fold'
tff(fact_7048_card_Oeq__fold,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),nat,finite_card(A),A4) = finite_fold(A,nat,aTP_Lamp_aea(A,fun(nat,nat)),zero_zero(nat),A4) ).

% card.eq_fold
tff(fact_7049_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
         => ( lattic5882676163264333800up_fin(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_7050_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
         => ( lattic7752659483105999362nf_fin(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_7051_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert(A,X),A4)) = X ) )
            & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert(A,X),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(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,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_7052_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
           => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( lattic7752659483105999362nf_fin(A,A4) = X ) )
              & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( lattic7752659483105999362nf_fin(A,A4) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(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,X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Inf_fin.remove
tff(fact_7053_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),insert(A,X),A4)) = X ) )
            & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),insert(A,X),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(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,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_7054_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
           => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( lattic5882676163264333800up_fin(A,A4) = X ) )
              & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( lattic5882676163264333800up_fin(A,A4) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(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,X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Sup_fin.remove
tff(fact_7055_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F2: fun(A,fun(B,B)),A4: set(A),X: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),S3))
       => ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
           => ( finite_fold(A,B,F2,Z,A4) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z,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,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_7056_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F2: fun(A,fun(B,B)),X: A,A4: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,X),A4)),S3))
       => ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),insert(A,X),A4)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z,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,X),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_7057_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S3: set(A),F2: fun(A,fun(B,B)),A4: set(A),Z: B,Y2: B,A3: A] :
      ( finite4664212375090638736ute_on(A,B,S3,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),S3))
       => ( finite_fold_graph(A,B,F2,Z,A4,Y2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
           => ? [Y6: B] :
                ( ( Y2 = aa(B,B,aa(A,fun(B,B),F2,A3),Y6) )
                & finite_fold_graph(A,B,F2,Z,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)))),Y6) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_7058_butlast__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),butlast(A),take(A,N,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_7059_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_7060_nth__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),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)),N) = aa(nat,A,nth(A,Xs),N) ) ) ).

% nth_butlast
tff(fact_7061_take__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( take(A,N,aa(list(A),list(A),butlast(A),Xs)) = take(A,N,Xs) ) ) ).

% take_butlast
tff(fact_7062_butlast__power,axiom,
    ! [A: $tType,N: 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))),N),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)),N),Xs) ).

% butlast_power
tff(fact_7063_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_7064_butlast__list__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),X: A] :
      ( ( ( 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),list_update(A,Xs,K,X)) = aa(list(A),list(A),butlast(A),Xs) ) )
      & ( ( 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),list_update(A,Xs,K,X)) = list_update(A,aa(list(A),list(A),butlast(A),Xs),K,X) ) ) ) ).

% butlast_list_update
tff(fact_7065_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat,S: A] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),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)),N) = infini527867602293511546merate(A,S3,N) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_7066_remove__def,axiom,
    ! [A: $tType,X: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),A4) = 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,X),bot_bot(set(A)))) ).

% remove_def
tff(fact_7067_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),M2: nat,N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),S3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S3,M2)),infini527867602293511546merate(A,S3,N)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ) ).

% enumerate_mono_iff
tff(fact_7068_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),M2: nat,N: nat] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(set(A),nat,finite_card(A),S3)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),S3)))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S3,M2)),infini527867602293511546merate(A,S3,N)))
              <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_7069_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),S3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S3,N)),infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)))) ) ) ).

% enumerate_step
tff(fact_7070_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M2: nat,N: nat,S3: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
         => ( ~ pp(aa(set(A),bool,finite_finite(A),S3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S3,M2)),infini527867602293511546merate(A,S3,N))) ) ) ) ).

% enumerate_mono
tff(fact_7071_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),S3)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),infini527867602293511546merate(A,S3,N)),S3)) ) ) ) ).

% finite_enumerate_in_set
tff(fact_7072_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),S: A] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S),S3))
           => ? [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(set(A),nat,finite_card(A),S3)))
                & ( infini527867602293511546merate(A,S3,N2) = S ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_7073_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y5: set(A)] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X6)))
             => ( infini527867602293511546merate(A,X6,I3) = infini527867602293511546merate(A,Y5,I3) ) )
         => ( pp(aa(set(A),bool,finite_finite(A),X6))
           => ( pp(aa(set(A),bool,finite_finite(A),Y5))
             => ( ( aa(set(A),nat,finite_card(A),X6) = aa(set(A),nat,finite_card(A),Y5) )
               => ( X6 = Y5 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_7074_minus__fold__remove,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4) = finite_fold(A,set(A),remove(A),B5,A4) ) ) ).

% minus_fold_remove
tff(fact_7075_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M2: nat,N: nat,S3: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
         => ( pp(aa(set(A),bool,finite_finite(A),S3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),S3)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S3,M2)),infini527867602293511546merate(A,S3,N))) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_7076_finite__le__enumerate,axiom,
    ! [S3: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(nat),nat,finite_card(nat),S3)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),infini527867602293511546merate(nat,S3,N))) ) ) ).

% finite_le_enumerate
tff(fact_7077_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S3,N)),infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)))) ) ) ) ).

% finite_enumerate_step
tff(fact_7078_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] : infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,infini527867602293511546merate(A,S3,zero_zero(nat))),bot_bot(set(A)))),N) ) ).

% enumerate_Suc'
tff(fact_7079_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y5: set(A)] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X6)))
             => ( infini527867602293511546merate(A,X6,I3) = infini527867602293511546merate(A,Y5,I3) ) )
         => ( pp(aa(set(A),bool,finite_finite(A),X6))
           => ( pp(aa(set(A),bool,finite_finite(A),Y5))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X6)),aa(set(A),nat,finite_card(A),Y5)))
               => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),Y5)) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_7080_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S3)))
           => ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_aeb(set(A),fun(nat,fun(A,bool)),S3),N)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_7081_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] : infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,ord_Least(A,aTP_Lamp_aec(set(A),fun(A,bool),S3))),bot_bot(set(A)))),N) ) ).

% enumerate_Suc
tff(fact_7082_Least__eq__0,axiom,
    ! [P: fun(nat,bool)] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ord_Least(nat,P) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_7083_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),ord_Least(A,P)))
         => ~ pp(aa(A,bool,P,K)) ) ) ).

% not_less_Least
tff(fact_7084_Least__Suc2,axiom,
    ! [P: fun(nat,bool),N: nat,Q: fun(nat,bool),M2: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( pp(aa(nat,bool,Q,M2))
       => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
         => ( ! [K2: nat] :
                ( pp(aa(nat,bool,P,aa(nat,nat,suc,K2)))
              <=> pp(aa(nat,bool,Q,K2)) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_7085_Least__Suc,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_wl(fun(nat,bool),fun(nat,bool),P))) ) ) ) ).

% Least_Suc
tff(fact_7086_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A)] : infini527867602293511546merate(A,S3,zero_zero(nat)) = ord_Least(A,aTP_Lamp_aec(set(A),fun(A,bool),S3)) ) ).

% enumerate_0
tff(fact_7087_Sup__real__def,axiom,
    ! [X6: set(real)] : aa(set(real),real,complete_Sup_Sup(real),X6) = ord_Least(real,aTP_Lamp_aed(set(real),fun(real,bool),X6)) ).

% Sup_real_def
tff(fact_7088_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),S3))
         => ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_aeb(set(A),fun(nat,fun(A,bool)),S3),N)) ) ) ) ).

% enumerate_Suc''
tff(fact_7089_Gcd__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite(A),A4))
           => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = finite_fold(A,A,gcd_gcd(A),zero_zero(A),A4) ) )
          & ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
           => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ) ).

% Gcd_fin.eq_fold
tff(fact_7090_independentD,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),T2: set(A),U: fun(A,real),V2: A] :
          ( ~ real_V358717886546972837endent(A,S)
         => ( pp(aa(set(A),bool,finite_finite(A),T2))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aee(fun(A,real),fun(A,A),U)),T2) = zero_zero(A) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),T2))
                 => ( aa(A,real,U,V2) = zero_zero(real) ) ) ) ) ) ) ) ).

% independentD
tff(fact_7091_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_7092_Gcd__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).

% Gcd_fin.infinite
tff(fact_7093_is__unit__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7094_dependent__single,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] :
          ( real_V358717886546972837endent(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A))))
        <=> ( X = zero_zero(A) ) ) ) ).

% dependent_single
tff(fact_7095_dependent__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A4: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A4))
         => real_V358717886546972837endent(A,A4) ) ) ).

% dependent_zero
tff(fact_7096_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(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),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_7097_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),insert(A,A3),A4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(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),bot_bot(set(A)))))) ) ).

% Gcd_fin.insert_remove
tff(fact_7098_independent__if__scalars__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ! [F3: fun(A,real),X2: A] :
                ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aee(fun(A,real),fun(A,A),F3)),A4) = zero_zero(A) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A4))
                 => ( aa(A,real,F3,X2) = zero_zero(real) ) ) )
           => ~ real_V358717886546972837endent(A,A4) ) ) ) ).

% independent_if_scalars_zero
tff(fact_7099_dependent__finite,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),S3))
         => ( real_V358717886546972837endent(A,S3)
          <=> ? [U3: fun(A,real)] :
                ( ? [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
                    & ( aa(A,real,U3,X4) != zero_zero(real) ) )
                & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aee(fun(A,real),fun(A,A),U3)),S3) = zero_zero(A) ) ) ) ) ) ).

% dependent_finite
tff(fact_7100_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = zero_zero(A) )
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A)))))
            & pp(aa(set(A),bool,finite_finite(A),A4)) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_7101_independentD__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B5: set(A),X6: fun(A,real),X: A] :
          ( ~ real_V358717886546972837endent(A,B5)
         => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_aef(fun(A,real),fun(A,bool),X6))))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_aef(fun(A,real),fun(A,bool),X6))),B5))
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aee(fun(A,real),fun(A,A),X6)),collect(A,aTP_Lamp_aef(fun(A,real),fun(A,bool),X6))) = zero_zero(A) )
               => ( aa(A,real,X6,X) = zero_zero(real) ) ) ) ) ) ) ).

% independentD_alt
tff(fact_7102_independent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B5: set(A)] :
          ( ~ real_V358717886546972837endent(A,B5)
        <=> ! [X7: fun(A,real)] :
              ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_aef(fun(A,real),fun(A,bool),X7))))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_aef(fun(A,real),fun(A,bool),X7))),B5))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aee(fun(A,real),fun(A,A),X7)),collect(A,aTP_Lamp_aef(fun(A,real),fun(A,bool),X7))) = zero_zero(A) )
                 => ! [X4: A] : aa(A,real,X7,X4) = zero_zero(real) ) ) ) ) ) ).

% independent_alt
tff(fact_7103_dependent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B5: set(A)] :
          ( real_V358717886546972837endent(A,B5)
        <=> ? [X7: fun(A,real)] :
              ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_aef(fun(A,real),fun(A,bool),X7))))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_aef(fun(A,real),fun(A,bool),X7))),B5))
              & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aee(fun(A,real),fun(A,A),X7)),collect(A,aTP_Lamp_aef(fun(A,real),fun(A,bool),X7))) = zero_zero(A) )
              & ? [X4: A] : aa(A,real,X7,X4) != zero_zero(real) ) ) ) ).

% dependent_alt
tff(fact_7104_independent__explicit__finite__subsets,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A4: set(A)] :
          ( ~ real_V358717886546972837endent(A,A4)
        <=> ! [S8: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S8),A4))
             => ( pp(aa(set(A),bool,finite_finite(A),S8))
               => ! [U3: fun(A,real)] :
                    ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aee(fun(A,real),fun(A,A),U3)),S8) = zero_zero(A) )
                   => ! [X4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S8))
                       => ( aa(A,real,U3,X4) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_finite_subsets
tff(fact_7105_independent__explicit__module,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( ~ real_V358717886546972837endent(A,S)
        <=> ! [T8: set(A),U3: fun(A,real),V5: A] :
              ( pp(aa(set(A),bool,finite_finite(A),T8))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T8),S))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aee(fun(A,real),fun(A,A),U3)),T8) = zero_zero(A) )
                 => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V5),T8))
                   => ( aa(A,real,U3,V5) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_module
tff(fact_7106_dependent__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( real_V358717886546972837endent(A,S)
        <=> ? [T8: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),T8))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T8),S))
              & ? [U3: fun(A,real)] :
                  ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aee(fun(A,real),fun(A,A),U3)),T8) = zero_zero(A) )
                  & ? [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),T8))
                      & ( aa(A,real,U3,X4) != zero_zero(real) ) ) ) ) ) ) ).

% dependent_explicit
tff(fact_7107_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),N: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,N))
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_7108_continuous__on__arcosh,axiom,
    ! [A4: set(real)] :
      ( pp(aa(set(real),bool,aa(set(real),fun(set(real),bool),ord_less_eq(set(real)),A4),set_ord_atLeast(real,one_one(real))))
     => topolo81223032696312382ous_on(real,real,A4,arcosh(real)) ) ).

% continuous_on_arcosh
tff(fact_7109_Compl__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),set_ord_lessThan(A,K)) = set_ord_atLeast(A,K) ) ).

% Compl_lessThan
tff(fact_7110_Compl__atLeast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),set_ord_atLeast(A,K)) = set_ord_lessThan(A,K) ) ).

% Compl_atLeast
tff(fact_7111_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),C2),set_ord_atMost(A,B2)) = set_ord_atLeast(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ).

% image_minus_const_AtMost
tff(fact_7112_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),C2),set_ord_atLeast(A,A3)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)) ) ).

% image_minus_const_atLeast
tff(fact_7113_image__uminus__atLeast,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : image(A,A,uminus_uminus(A),set_ord_atLeast(A,X)) = set_ord_atMost(A,aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atLeast
tff(fact_7114_image__uminus__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : image(A,A,uminus_uminus(A),set_ord_atMost(A,X)) = set_ord_atLeast(A,aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atMost
tff(fact_7115_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : pp(bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat))) ) ).

% possible_bit_0
tff(fact_7116_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,A3)),set_ord_greaterThan(A,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_7117_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7118_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N),bit_se6407376104438227557le_bit(A,type2(A),N)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ).

% drop_bit_exp_eq
tff(fact_7119_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% bit_minus_2_iff
tff(fact_7120_atLeast__0,axiom,
    set_ord_atLeast(nat,zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_7121_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_7122_of__nat__CHAR,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),semiri4206861660011772517g_char(A,type2(A))) = zero_zero(A) ) ) ).

% of_nat_CHAR
tff(fact_7123_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),one_one(A))),N))
        <=> pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).

% bit_minus_1_iff
tff(fact_7124_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = set_ord_greaterThan(nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_7125_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,M2)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ) ).

% bit_mask_iff
tff(fact_7126_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),semiri4206861660011772517g_char(A,type2(A))),N)) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_7127_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X2: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X2) = zero_zero(A) )
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),C2),X2)) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_7128_bit__of__nat__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M2)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M2),N)) ) ) ) ).

% bit_of_nat_iff
tff(fact_7129_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A3)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N)) ) ) ) ).

% bit_minus_iff
tff(fact_7130_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A))))
      <=> ? [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
            & ( aa(nat,A,semiring_1_of_nat(A),N5) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7131_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C2))
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X2))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),C2))
                   => ( aa(nat,A,semiring_1_of_nat(A),X2) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7132_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
           => ( aa(nat,A,semiring_1_of_nat(A),N5) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7133_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4730199178511100633sh_bit(A,M2),A3)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% bit_push_bit_iff
tff(fact_7134_atLeast__Suc,axiom,
    ! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atLeast(nat,K)),aa(set(nat),set(nat),insert(nat,K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_7135_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
        <=> ~ pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).

% fold_possible_bit
tff(fact_7136_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),one_one(nat)))
            & ( N = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_7137_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7138_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,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))),M2)),one_one(A))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7139_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
            & ( N != zero_zero(nat) )
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ).

% bit_double_iff
tff(fact_7140_last__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,X: A] :
      ( ( Xs != nil(A) )
     => ( ( ( K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
         => ( last(A,list_update(A,Xs,K,X)) = X ) )
        & ( ( K != aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
         => ( last(A,list_update(A,Xs,K,X)) = last(A,Xs) ) ) ) ) ).

% last_list_update
tff(fact_7141_minus__coset__filter,axiom,
    ! [A: $tType,A4: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),coset(A,Xs)) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_a(set(A),fun(A,bool),A4),Xs)) ).

% minus_coset_filter
tff(fact_7142_last__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( last(A,replicate(A,N,X)) = X ) ) ).

% last_replicate
tff(fact_7143_last__upt,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
     => ( last(nat,upt(I,J2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),one_one(nat)) ) ) ).

% last_upt
tff(fact_7144_last__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( last(A,drop(A,N,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_7145_last__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
         => ( last(product_prod(A,B),zip(A,B,Xs,Ys2)) = aa(B,product_prod(A,B),product_Pair(A,B,last(A,Xs)),last(B,Ys2)) ) ) ) ) ).

% last_zip
tff(fact_7146_coset__def,axiom,
    ! [A: $tType,Xs: list(A)] : coset(A,Xs) = aa(set(A),set(A),uminus_uminus(set(A)),aa(list(A),set(A),set2(A),Xs)) ).

% coset_def
tff(fact_7147_compl__coset,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),set(A),uminus_uminus(set(A)),coset(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% compl_coset
tff(fact_7148_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_7149_flat__lub__def,axiom,
    ! [A: $tType,A4: set(A),B2: A] :
      ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A4) = B2 ) )
      & ( ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A4) = the(A,aa(A,fun(A,bool),aTP_Lamp_aeg(set(A),fun(A,fun(A,bool)),A4),B2)) ) ) ) ).

% flat_lub_def
tff(fact_7150_in__measures_I2_J,axiom,
    ! [A: $tType,X: A,Y2: A,F2: fun(A,nat),Fs: list(fun(A,nat))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),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))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y2)))
        | ( ( aa(A,nat,F2,X) = aa(A,nat,F2,Y2) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),measures(A,Fs))) ) ) ) ).

% in_measures(2)
tff(fact_7151_measures__less,axiom,
    ! [A: $tType,F2: fun(A,nat),X: A,Y2: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y2)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),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_7152_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(bool,fun(bool,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),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(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_aeh(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),F1),F22),X13)),X14),aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),F1),F22),X14)) ).

% VEBT.simps(7)
tff(fact_7153_and__not__num_Opelims,axiom,
    ! [X: num,Xa: num,Y2: option(num)] :
      ( ( bit_and_not_num(X,Xa) = Y2 )
     => ( accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,X),Xa))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y2 = none(num) )
               => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa = aa(num,num,bit0,N2) )
                 => ( ( Y2 = aa(num,option(num),some(num),one2) )
                   => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,N2))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa = aa(num,num,bit1,N2) )
                   => ( ( Y2 = none(num) )
                     => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ( ( Xa = one2 )
                     => ( ( Y2 = aa(num,option(num),some(num),aa(num,num,bit0,M3)) )
                       => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M3)),one2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit0,N2) )
                         => ( ( Y2 = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M3,N2)) )
                           => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M3)),aa(num,num,bit0,N2))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit0,M3) )
                       => ! [N2: num] :
                            ( ( Xa = aa(num,num,bit1,N2) )
                           => ( ( Y2 = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M3,N2)) )
                             => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M3)),aa(num,num,bit1,N2))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ( ( Xa = one2 )
                           => ( ( Y2 = aa(num,option(num),some(num),aa(num,num,bit0,M3)) )
                             => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M3)),one2)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit0,N2) )
                               => ( ( Y2 = 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_nj(num,option(num))),bit_and_not_num(M3,N2)) )
                                 => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M3)),aa(num,num,bit0,N2))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa(num,num,bit1,M3) )
                             => ! [N2: num] :
                                  ( ( Xa = aa(num,num,bit1,N2) )
                                 => ( ( Y2 = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M3,N2)) )
                                   => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M3)),aa(num,num,bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_7154_VEBT_Osimps_I8_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(bool,fun(bool,A)),X21: bool,X22: bool] : aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),F1),F22),vEBT_Leaf(X21,X22)) = aa(bool,A,aa(bool,fun(bool,A),F22,X21),X22) ).

% VEBT.simps(8)
tff(fact_7155_and__num_Opelims,axiom,
    ! [X: num,Xa: num,Y2: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa) = Y2 )
     => ( accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,X),Xa))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y2 = aa(num,option(num),some(num),one2) )
               => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa = aa(num,num,bit0,N2) )
                 => ( ( Y2 = none(num) )
                   => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,N2))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa = aa(num,num,bit1,N2) )
                   => ( ( Y2 = aa(num,option(num),some(num),one2) )
                     => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ( ( Xa = one2 )
                     => ( ( Y2 = none(num) )
                       => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M3)),one2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit0,N2) )
                         => ( ( Y2 = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M3,N2)) )
                           => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M3)),aa(num,num,bit0,N2))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit0,M3) )
                       => ! [N2: num] :
                            ( ( Xa = aa(num,num,bit1,N2) )
                           => ( ( Y2 = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M3,N2)) )
                             => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M3)),aa(num,num,bit1,N2))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ( ( Xa = one2 )
                           => ( ( Y2 = aa(num,option(num),some(num),one2) )
                             => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M3)),one2)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit0,N2) )
                               => ( ( Y2 = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M3,N2)) )
                                 => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M3)),aa(num,num,bit0,N2))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa(num,num,bit1,M3) )
                             => ! [N2: num] :
                                  ( ( Xa = aa(num,num,bit1,N2) )
                                 => ( ( Y2 = 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_nj(num,option(num))),bit_un7362597486090784418nd_num(M3,N2)) )
                                   => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M3)),aa(num,num,bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_7156_xor__num_Opelims,axiom,
    ! [X: num,Xa: num,Y2: option(num)] :
      ( ( bit_un2480387367778600638or_num(X,Xa) = Y2 )
     => ( accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,X),Xa))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y2 = none(num) )
               => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa = aa(num,num,bit0,N2) )
                 => ( ( Y2 = aa(num,option(num),some(num),aa(num,num,bit1,N2)) )
                   => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,N2))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa = aa(num,num,bit1,N2) )
                   => ( ( Y2 = aa(num,option(num),some(num),aa(num,num,bit0,N2)) )
                     => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ( ( Xa = one2 )
                     => ( ( Y2 = aa(num,option(num),some(num),aa(num,num,bit1,M3)) )
                       => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M3)),one2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit0,N2) )
                         => ( ( Y2 = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un2480387367778600638or_num(M3,N2)) )
                           => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M3)),aa(num,num,bit0,N2))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit0,M3) )
                       => ! [N2: num] :
                            ( ( Xa = aa(num,num,bit1,N2) )
                           => ( ( Y2 = 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_un2480387367778600638or_num(M3,N2))) )
                             => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M3)),aa(num,num,bit1,N2))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ( ( Xa = one2 )
                           => ( ( Y2 = aa(num,option(num),some(num),aa(num,num,bit0,M3)) )
                             => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M3)),one2)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit0,N2) )
                               => ( ( Y2 = 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_un2480387367778600638or_num(M3,N2))) )
                                 => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M3)),aa(num,num,bit0,N2))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa(num,num,bit1,M3) )
                             => ! [N2: num] :
                                  ( ( Xa = aa(num,num,bit1,N2) )
                                 => ( ( Y2 = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un2480387367778600638or_num(M3,N2)) )
                                   => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M3)),aa(num,num,bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_7157_pair__lessI2,axiom,
    ! [A3: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
       => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),T2))),fun_pair_less)) ) ) ).

% pair_lessI2
tff(fact_7158_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,bool)),B5: fun(C,fun(D,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A4,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),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)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),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)))
             => pp(aa(fun(fun(D,B),fun(B,fun(list(D),B))),bool,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),bool),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,B5,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,B5),A4))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B))) ) ) ) ) ).

% horner_sum_transfer
tff(fact_7159_pair__less__iff1,axiom,
    ! [X: nat,Y2: nat,Z: nat] :
      ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Y2)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Z))),fun_pair_less))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y2),Z)) ) ).

% pair_less_iff1
tff(fact_7160_list__all2__lengthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) ) ) ).

% list_all2_lengthD
tff(fact_7161_length__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),nat),bool,aa(fun(list(A),nat),fun(fun(list(B),nat),bool),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_7162_list__all2__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),P: fun(A,fun(B,bool)),Us: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),append(A,Xs,Us)),append(B,Ys2,Vs)))
      <=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Us),Vs)) ) ) ) ).

% list_all2_append
tff(fact_7163_list__all2__append1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(A),Zs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),append(A,Xs,Ys2)),Zs))
    <=> ? [Us2: list(B),Vs2: list(B)] :
          ( ( Zs = append(B,Us2,Vs2) )
          & ( aa(list(B),nat,size_size(list(B)),Us2) = aa(list(A),nat,size_size(list(A)),Xs) )
          & ( aa(list(B),nat,size_size(list(B)),Vs2) = aa(list(A),nat,size_size(list(A)),Ys2) )
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Us2))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Ys2),Vs2)) ) ) ).

% list_all2_append1
tff(fact_7164_list__all2__append2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B),Zs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),append(B,Ys2,Zs)))
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Xs = append(A,Us2,Vs2) )
          & ( aa(list(A),nat,size_size(list(A)),Us2) = aa(list(B),nat,size_size(list(B)),Ys2) )
          & ( aa(list(A),nat,size_size(list(A)),Vs2) = aa(list(B),nat,size_size(list(B)),Zs) )
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Us2),Ys2))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Vs2),Zs)) ) ) ).

% list_all2_append2
tff(fact_7165_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B),P2: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P2),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P2)),aa(nat,B,nth(B,Ys2),P2))) ) ) ).

% list_all2_nthD
tff(fact_7166_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B),P2: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P2),aa(list(B),nat,size_size(list(B)),Ys2)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P2)),aa(nat,B,nth(B,Ys2),P2))) ) ) ).

% list_all2_nthD2
tff(fact_7167_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),A3)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,A3),N2)),aa(nat,B,nth(B,B2),N2))) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),A3),B2)) ) ) ).

% list_all2_all_nthI
tff(fact_7168_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Ys2),I2))) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_7169_pair__lessI1,axiom,
    ! [A3: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),T2))),fun_pair_less)) ) ).

% pair_lessI1
tff(fact_7170_list__all2I,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
      ( ! [X2: product_prod(A,B)] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,A3,B2))))
         => pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,P),X2)) )
     => ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),A3),B2)) ) ) ).

% list_all2I
tff(fact_7171_sum__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & monoid_add(A) )
     => ! [A4: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A4,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),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)))
           => pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),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_7172_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
           => pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,P),X4)) ) ) ) ).

% list_all2_iff
tff(fact_7173_prod__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_mult(B)
        & monoid_mult(A) )
     => ! [A4: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A4,one_one(A)),one_one(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),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)))
           => pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),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_7174_pair__leqI1,axiom,
    ! [A3: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),T2))),fun_pair_leq)) ) ).

% pair_leqI1
tff(fact_7175_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_7176_prod__list__zero__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Xs: list(A)] :
          ( ( aa(list(A),A,groups5270119922927024881d_list(A),Xs) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% prod_list_zero_iff
tff(fact_7177_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_7178_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_aei(A,fun(A,bool)),aTP_Lamp_aej(A,fun(A,bool)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_7179_dropWhile__nth,axiom,
    ! [A: $tType,J2: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))))
     => ( aa(nat,A,nth(A,dropWhile(A,P,Xs)),J2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_7180_length__dropWhile__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_7181_dropWhile__eq__drop,axiom,
    ! [A: $tType,P: fun(A,bool),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_7182_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_lm(nat,fun(nat,bool)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_7183_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_7184_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_lb(nat,fun(nat,bool)),aTP_Lamp_cv(nat,fun(nat,bool)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_7185_find__dropWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : find(A,P,Xs) = case_list(option(A),A,none(A),aTP_Lamp_aek(A,fun(list(A),option(A))),dropWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P),Xs)) ).

% find_dropWhile
tff(fact_7186_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_7187_vimage__if,axiom,
    ! [B: $tType,A: $tType,C2: B,A4: set(B),D2: B,B5: set(A)] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A4))
       => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A4))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ael(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B5),A4) = top_top(set(A)) ) )
          & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A4))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ael(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B5),A4) = B5 ) ) ) )
      & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A4))
       => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A4))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ael(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B5),A4) = aa(set(A),set(A),uminus_uminus(set(A)),B5) ) )
          & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A4))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ael(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B5),A4) = bot_bot(set(A)) ) ) ) ) ) ).

% vimage_if
tff(fact_7188_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B)] : vimage(A,B,F2,aa(set(B),set(B),uminus_uminus(set(B)),A4)) = aa(set(A),set(A),uminus_uminus(set(A)),vimage(A,B,F2,A4)) ).

% vimage_Compl
tff(fact_7189_vimage__Diff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B),B5: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),vimage(A,B,F2,A4)),vimage(A,B,F2,B5)) ).

% vimage_Diff
tff(fact_7190_finite__vimage__Suc__iff,axiom,
    ! [F4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),vimage(nat,nat,suc,F4)))
    <=> pp(aa(set(nat),bool,finite_finite(nat),F4)) ) ).

% finite_vimage_Suc_iff
tff(fact_7191_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A4: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),A4)) = aa(set(nat),set(nat),insert(nat,N),vimage(nat,nat,suc,A4)) ).

% vimage_Suc_insert_Suc
tff(fact_7192_vimage__Suc__insert__0,axiom,
    ! [A4: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),insert(nat,zero_zero(nat)),A4)) = vimage(nat,nat,suc,A4) ).

% vimage_Suc_insert_0
tff(fact_7193_set__decode__div__2,axiom,
    ! [X: nat] : nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = vimage(nat,nat,suc,nat_set_decode(X)) ).

% set_decode_div_2
tff(fact_7194_extract__def,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : extract(A,P,Xs) = case_list(option(product_prod(list(A),product_prod(A,list(A)))),A,none(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_aem(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs),dropWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P),Xs)) ).

% extract_def
tff(fact_7195_euclidean__size__times__nonunit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,B2)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_7196_euclidean__size__of__nat,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : euclid6346220572633701492n_size(A,aa(nat,A,semiring_1_of_nat(A),N)) = N ) ).

% euclidean_size_of_nat
tff(fact_7197_euclidean__size__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( ( euclid6346220572633701492n_size(A,B2) = zero_zero(nat) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% euclidean_size_eq_0_iff
tff(fact_7198_size__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ( euclid6346220572633701492n_size(A,zero_zero(A)) = zero_zero(nat) ) ) ).

% size_0
tff(fact_7199_euclidean__size__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( euclid6346220572633701492n_size(A,one_one(A)) = one_one(nat) ) ) ).

% euclidean_size_1
tff(fact_7200_euclidean__size__greater__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),euclid6346220572633701492n_size(A,B2)))
        <=> ( B2 != zero_zero(A) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_7201_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,B2) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_7202_euclidean__size__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,one_one(A)) ) ) ) ).

% euclidean_size_unit
tff(fact_7203_extract__Nil__code,axiom,
    ! [A: $tType,P: fun(A,bool)] : extract(A,P,nil(A)) = none(product_prod(list(A),product_prod(A,list(A)))) ).

% extract_Nil_code
tff(fact_7204_extract__None__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( extract(A,P,Xs) = none(product_prod(list(A),product_prod(A,list(A)))) )
    <=> ~ ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(A,bool,P,X4)) ) ) ).

% extract_None_iff
tff(fact_7205_unit__iff__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
        <=> ( ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,one_one(A)) )
            & ( A3 != zero_zero(A) ) ) ) ) ).

% unit_iff_euclidean_size
tff(fact_7206_size__mult__mono,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ).

% size_mult_mono
tff(fact_7207_size__mult__mono_H,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))) ) ) ).

% size_mult_mono'
tff(fact_7208_euclidean__size__times__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = euclid6346220572633701492n_size(A,B2) ) ) ) ).

% euclidean_size_times_unit
tff(fact_7209_dvd__proper__imp__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( ( B2 != zero_zero(A) )
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2))) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_7210_dvd__imp__size__le,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( ( B2 != zero_zero(A) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2))) ) ) ) ).

% dvd_imp_size_le
tff(fact_7211_mod__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,modulo_modulo(A,A3,B2))),euclid6346220572633701492n_size(A,B2))) ) ) ).

% mod_size_less
tff(fact_7212_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) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),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_7213_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q5: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R)),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),Q5),B2)),R) = A3 )
               => ( modulo_modulo(A,A3,B2) = R ) ) ) ) ) ) ).

% mod_eqI
tff(fact_7214_abs__division__segment,axiom,
    ! [K: int] : aa(int,int,abs_abs(int),euclid7384307370059645450egment(int,K)) = one_one(int) ).

% abs_division_segment
tff(fact_7215_division__segment__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( euclid7384307370059645450egment(A,one_one(A)) = one_one(A) ) ) ).

% division_segment_1
tff(fact_7216_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_7217_division__segment__of__nat,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : euclid7384307370059645450egment(A,aa(nat,A,semiring_1_of_nat(A),N)) = one_one(A) ) ).

% division_segment_of_nat
tff(fact_7218_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),euclid6346220572633701492n_size(A,A3))) = A3 ) ).

% division_segment_euclidean_size
tff(fact_7219_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_7220_division__segment__nat__def,axiom,
    ! [N: nat] : euclid7384307370059645450egment(nat,N) = one_one(nat) ).

% division_segment_nat_def
tff(fact_7221_division__segment__not__0,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A] : euclid7384307370059645450egment(A,A3) != zero_zero(A) ) ).

% division_segment_not_0
tff(fact_7222_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( euclid7384307370059645450egment(int,K) = aa(int,int,sgn_sgn(int),K) ) ) ).

% division_segment_eq_sgn
tff(fact_7223_is__unit__division__segment,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),euclid7384307370059645450egment(A,A3)),one_one(A))) ) ).

% is_unit_division_segment
tff(fact_7224_division__segment__mod,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( euclid7384307370059645450egment(A,modulo_modulo(A,A3,B2)) = euclid7384307370059645450egment(A,B2) ) ) ) ) ).

% division_segment_mod
tff(fact_7225_of__nat__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] : aa(nat,A,semiring_1_of_nat(A),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_7226_division__segment__int__def,axiom,
    ! [K: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
       => ( euclid7384307370059645450egment(int,K) = one_one(int) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
       => ( euclid7384307370059645450egment(int,K) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% division_segment_int_def
tff(fact_7227_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) )
          <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2)))
              | ( B2 = zero_zero(A) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_7228_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q5: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R)),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),Q5),B2)),R)),B2) = Q5 ) ) ) ) ) ).

% div_bounded
tff(fact_7229_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q5: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R)),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),Q5),B2)),R) = A3 )
               => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q5 ) ) ) ) ) ) ).

% div_eqI
tff(fact_7230_sorted__wrt__iff__nth__Suc__transp,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( transp(A,P)
     => ( sorted_wrt(A,P,Xs)
      <=> ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I2)))) ) ) ) ).

% sorted_wrt_iff_nth_Suc_transp
tff(fact_7231_mlex__eq,axiom,
    ! [A: $tType,F2: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F2,R2) = collect(product_prod(A,A),product_case_prod(A,A,bool,aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_aen(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F2),R2))) ).

% mlex_eq
tff(fact_7232_transp__realrel,axiom,
    transp(fun(nat,rat),realrel) ).

% transp_realrel
tff(fact_7233_mlex__less,axiom,
    ! [A: $tType,F2: fun(A,nat),X: A,Y2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y2)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),mlex_prod(A,F2,R2))) ) ).

% mlex_less
tff(fact_7234_mlex__iff,axiom,
    ! [A: $tType,X: A,Y2: A,F2: fun(A,nat),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),mlex_prod(A,F2,R2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y2)))
        | ( ( aa(A,nat,F2,X) = aa(A,nat,F2,Y2) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),R2)) ) ) ) ).

% mlex_iff
tff(fact_7235_in__measure,axiom,
    ! [A: $tType,X: A,Y2: A,F2: fun(A,nat)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),measure(A,F2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y2))) ) ).

% in_measure
tff(fact_7236_transp__gr,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_aeo(A,fun(A,bool))) ) ).

% transp_gr
tff(fact_7237_transp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less(A)) ) ).

% transp_less
tff(fact_7238_pred__nat__def,axiom,
    pred_nat = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_aep(nat,fun(nat,bool)))) ).

% pred_nat_def
tff(fact_7239_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
        <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2)),lexord(A,collect(product_prod(A,A),product_case_prod(A,A,bool,ord_less(A)))))) ) ) ).

% lexordp_conv_lexord
tff(fact_7240_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A),Y2: A,Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
              & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2)) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_7241_lexordp__irreflexive,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] :
          ( ! [X2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),X2))
         => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Xs)) ) ) ).

% lexordp_irreflexive
tff(fact_7242_lexordp__append__leftD,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A),Us: list(A),Vs: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),append(A,Xs,Us)),append(A,Xs,Vs)))
         => ( ! [A6: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A6),A6))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us),Vs)) ) ) ) ).

% lexordp_append_leftD
tff(fact_7243_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y2: A,Xs: list(A),Ys2: list(A)] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
           => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_7244_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y2: A,Xs: list(A),Ys2: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))) ) ) ).

% lexordp.Cons
tff(fact_7245_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A),P: fun(list(A),fun(list(A),bool))] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
         => ( ! [Y3: A,Ys5: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)))
           => ( ! [X2: A,Xs2: list(A),Y3: A,Ys5: list(A)] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3))
                 => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5))) )
             => ( ! [X2: A,Xs2: list(A),Ys5: list(A)] :
                    ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys5))
                   => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),Ys5))
                     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys5))) ) )
               => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs),Ys2)) ) ) ) ) ) ).

% lexordp_induct
tff(fact_7246_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
         => ( ( ( Xs = nil(A) )
             => ! [Y3: A,Ys4: list(A)] : Ys2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
           => ( ! [X2: A] :
                  ( ? [Xs4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs4)
                 => ! [Y3: A] :
                      ( ? [Ys4: list(A)] : Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4)
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3)) ) )
             => ~ ! [X2: A,Xs4: list(A)] :
                    ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs4) )
                   => ! [Ys4: list(A)] :
                        ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys4) )
                       => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs4),Ys4)) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_7247_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
        <=> ( ? [Y4: A,Ys3: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) ) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys3: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4)) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys3: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X4))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs3),Ys3)) ) ) ) ) ).

% lexordp.simps
tff(fact_7248_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
         => ( ( ( A1 = nil(A) )
             => ! [Y3: A,Ys5: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
           => ( ! [X2: A] :
                  ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)
                 => ! [Y3: A] :
                      ( ? [Ys5: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3)) ) )
             => ~ ! [X2: A,Y3: A,Xs2: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
                   => ! [Ys5: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
                       => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3))
                         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X2))
                           => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys5)) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_7249_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y2: A,Us: list(A),Xs: list(A),Ys2: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),append(A,Us,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))),append(A,Us,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))) ) ) ).

% lexordp_append_left_rightI
tff(fact_7250_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
        <=> ( ? [X4: A,Vs2: list(A)] : Ys2 = append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Vs2))
            | ? [Us2: list(A),A5: A,B3: A,Vs2: list(A),Ws3: list(A)] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A5),B3))
                & ( Xs = append(A,Us2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),Vs2)) )
                & ( Ys2 = append(A,Us2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B3),Ws3)) ) ) ) ) ) ).

% lexordp_iff
tff(fact_7251_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_acp(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ) ).

% ord_class.lexordp_def
tff(fact_7252_less__enat__def,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),M2),N))
    <=> pp(extended_case_enat(bool,aTP_Lamp_aeq(extended_enat,fun(nat,bool),N),fFalse,M2)) ) ).

% less_enat_def
tff(fact_7253_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),N: nat] :
          ( 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,N)),F2)) = complete_lattice_lfp(A,F2) ) ) ) ).

% lfp_funpow
tff(fact_7254_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( 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_7255_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,product_case_prod(int,int,int,aTP_Lamp_aer(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_7256_suminf__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),image(nat,nat,G,top_top(set(nat)))))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => ( suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aes(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2)) = suminf(A,F2) ) ) ) ) ).

% suminf_mono_reindex
tff(fact_7257_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_strict_mono(nat,A,F2)
        <=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N5)),aa(nat,A,F2,aa(nat,nat,suc,N5)))) ) ) ).

% strict_mono_Suc_iff
tff(fact_7258_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y2: A] :
          ( order_strict_mono(A,B,F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y2))) ) ) ) ).

% strict_monoD
tff(fact_7259_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X2: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y3))) )
         => order_strict_mono(A,B,F2) ) ) ).

% strict_monoI
tff(fact_7260_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
        <=> ! [X4: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y4))) ) ) ) ).

% strict_mono_def
tff(fact_7261_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y2: A] :
          ( order_strict_mono(A,B,F2)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2)) ) ) ) ).

% strict_mono_less
tff(fact_7262_strict__mono__imp__increasing,axiom,
    ! [F2: fun(nat,nat),N: nat] :
      ( order_strict_mono(nat,nat,F2)
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,F2,N))) ) ).

% strict_mono_imp_increasing
tff(fact_7263_rat__floor__code,axiom,
    ! [P2: rat] : archim6421214686448440834_floor(rat,P2) = aa(product_prod(int,int),int,product_case_prod(int,int,int,divide_divide(int)),quotient_of(P2)) ).

% rat_floor_code
tff(fact_7264_summable__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),image(nat,nat,G,top_top(set(nat)))))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aet(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2))
            <=> summable(A,F2) ) ) ) ) ).

% summable_mono_reindex
tff(fact_7265_sums__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A),C2: A] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),image(nat,nat,G,top_top(set(nat)))))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aet(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2),C2)
            <=> sums(A,F2,C2) ) ) ) ) ).

% sums_mono_reindex
tff(fact_7266_prod__decode__triangle__add,axiom,
    ! [K: nat,M2: nat] : aa(nat,product_prod(nat,nat),nat_prod_decode,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M2)) = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M2) ).

% prod_decode_triangle_add
tff(fact_7267_map__comp__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,option(C)),G: fun(A,option(B)),X3: A] : map_comp(B,C,A,F2,G,X3) = aa(option(B),option(C),aa(fun(B,option(C)),fun(option(B),option(C)),aa(option(C),fun(fun(B,option(C)),fun(option(B),option(C))),case_option(option(C),B),none(C)),F2),aa(A,option(B),G,X3)) ).

% map_comp_def
tff(fact_7268_prod__decode__eq,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode,X) = aa(nat,product_prod(nat,nat),nat_prod_decode,Y2) )
    <=> ( X = Y2 ) ) ).

% prod_decode_eq
tff(fact_7269_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_7270_prod__decode__inverse,axiom,
    ! [N: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),nat_prod_decode,N)) = N ).

% prod_decode_inverse
tff(fact_7271_prod__encode__inverse,axiom,
    ! [X: product_prod(nat,nat)] : aa(nat,product_prod(nat,nat),nat_prod_decode,aa(product_prod(nat,nat),nat,nat_prod_encode,X)) = X ).

% prod_encode_inverse
tff(fact_7272_map__comp__empty_I2_J,axiom,
    ! [C: $tType,B: $tType,D: $tType,M2: fun(C,option(B)),X3: C] : map_comp(B,D,C,aTP_Lamp_aeu(B,option(D)),M2,X3) = none(D) ).

% map_comp_empty(2)
tff(fact_7273_map__comp__empty_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,M2: fun(C,option(B)),X3: A] : map_comp(C,B,A,M2,aTP_Lamp_nr(A,option(C)),X3) = none(B) ).

% map_comp_empty(1)
tff(fact_7274_tanh__real__strict__mono,axiom,
    order_strict_mono(real,real,tanh(real)) ).

% tanh_real_strict_mono
tff(fact_7275_sinh__real__strict__mono,axiom,
    order_strict_mono(real,real,sinh(real)) ).

% sinh_real_strict_mono
tff(fact_7276_inj__prod__decode,axiom,
    ! [A4: set(nat)] : inj_on(nat,product_prod(nat,nat),nat_prod_decode,A4) ).

% inj_prod_decode
tff(fact_7277_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero(nat)) ).

% prod_decode_def
tff(fact_7278_bij__prod__decode,axiom,
    bij_betw(nat,product_prod(nat,nat),nat_prod_decode,top_top(set(nat)),top_top(set(product_prod(nat,nat)))) ).

% bij_prod_decode
tff(fact_7279_surj__prod__decode,axiom,
    image(nat,product_prod(nat,nat),nat_prod_decode,top_top(set(nat))) = top_top(set(product_prod(nat,nat))) ).

% surj_prod_decode
tff(fact_7280_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) )
        | ? [K9: B] :
            ( ( aa(C,option(B),M22,K) = aa(B,option(B),some(B),K9) )
            & ( aa(B,option(A),M1,K9) = none(A) ) ) ) ) ).

% map_comp_None_iff
tff(fact_7281_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun(nat,bool)] :
      ( accp(nat,nat_list_decode_rel,A0)
     => ( ( accp(nat,nat_list_decode_rel,zero_zero(nat))
         => pp(aa(nat,bool,P,zero_zero(nat))) )
       => ( ! [N2: nat] :
              ( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N2))
             => ( ! [X3: nat,Y: nat] :
                    ( ( aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),Y) = aa(nat,product_prod(nat,nat),nat_prod_decode,N2) )
                   => pp(aa(nat,bool,P,Y)) )
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,A0)) ) ) ) ).

% list_decode.pinduct
tff(fact_7282_list__decode_Oelims,axiom,
    ! [X: nat,Y2: list(nat)] :
      ( ( aa(nat,list(nat),nat_list_decode,X) = Y2 )
     => ( ( ( X = zero_zero(nat) )
         => ( Y2 != nil(nat) ) )
       => ~ ! [N2: nat] :
              ( ( X = aa(nat,nat,suc,N2) )
             => ( Y2 != aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_aev(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_7283_list__decode__inverse,axiom,
    ! [N: nat] : aa(list(nat),nat,nat_list_encode,aa(nat,list(nat),nat_list_decode,N)) = N ).

% list_decode_inverse
tff(fact_7284_list__encode__inverse,axiom,
    ! [X: list(nat)] : aa(nat,list(nat),nat_list_decode,aa(list(nat),nat,nat_list_encode,X)) = X ).

% list_encode_inverse
tff(fact_7285_list__decode_Opsimps_I1_J,axiom,
    ( accp(nat,nat_list_decode_rel,zero_zero(nat))
   => ( aa(nat,list(nat),nat_list_decode,zero_zero(nat)) = nil(nat) ) ) ).

% list_decode.psimps(1)
tff(fact_7286_inj__list__decode,axiom,
    ! [A4: set(nat)] : inj_on(nat,list(nat),nat_list_decode,A4) ).

% inj_list_decode
tff(fact_7287_list__decode__eq,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa(nat,list(nat),nat_list_decode,X) = aa(nat,list(nat),nat_list_decode,Y2) )
    <=> ( X = Y2 ) ) ).

% list_decode_eq
tff(fact_7288_list__decode_Osimps_I1_J,axiom,
    aa(nat,list(nat),nat_list_decode,zero_zero(nat)) = nil(nat) ).

% list_decode.simps(1)
tff(fact_7289_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N))
     => ( aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_aev(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_7290_bij__list__decode,axiom,
    bij_betw(nat,list(nat),nat_list_decode,top_top(set(nat)),top_top(set(list(nat)))) ).

% bij_list_decode
tff(fact_7291_surj__list__decode,axiom,
    image(nat,list(nat),nat_list_decode,top_top(set(nat))) = top_top(set(list(nat))) ).

% surj_list_decode
tff(fact_7292_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] : aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_aev(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ).

% list_decode.simps(2)
tff(fact_7293_list__decode_Opelims,axiom,
    ! [X: nat,Y2: list(nat)] :
      ( ( aa(nat,list(nat),nat_list_decode,X) = Y2 )
     => ( accp(nat,nat_list_decode_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y2 = nil(nat) )
             => ~ accp(nat,nat_list_decode_rel,zero_zero(nat)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa(nat,nat,suc,N2) )
               => ( ( Y2 = aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_aev(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N2)) )
                 => ~ accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N2)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_7294_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_aew(A,fun(A,bool)),aTP_Lamp_aex(A,fun(A,bool))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_7295_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_7296_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_7297_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_7298_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_aey(A,fun(A,bool)),aTP_Lamp_aez(A,fun(A,bool))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_7299_bounded__quasi__semilattice__set_Oremove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
       => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A4) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),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)))))) ) ) ) ).

% bounded_quasi_semilattice_set.remove
tff(fact_7300_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),insert(A,A3),A4)) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),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)))))) ) ) ).

% bounded_quasi_semilattice_set.insert_remove
tff(fact_7301_gen__length__def,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,gen_length(A,N),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)) ).

% gen_length_def
tff(fact_7302_compute__powr__real,axiom,
    ! [B2: real,I: real] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),zero_zero(real)))
       => ( aa(real,real,aa(real,fun(real,real),powr_real,B2),I) = abort(real,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,zero_zero(literal)))))))))))))))))))))))))))))))),aa(real,fun(product_unit,real),aTP_Lamp_afa(real,fun(real,fun(product_unit,real)),B2),I)) ) )
      & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),zero_zero(real)))
       => ( ( ( aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,I)) = I )
           => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),I))
               => ( aa(real,real,aa(real,fun(real,real),powr_real,B2),I) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(int,nat,nat2,archim6421214686448440834_floor(real,I))) ) )
              & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),I))
               => ( aa(real,real,aa(real,fun(real,real),powr_real,B2),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))))) ) ) ) )
          & ( ( aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,I)) != I )
           => ( aa(real,real,aa(real,fun(real,real),powr_real,B2),I) = abort(real,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fTrue,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,zero_zero(literal)))))))))))))))))))))))))))))))))))),aa(real,fun(product_unit,real),aTP_Lamp_afa(real,fun(real,fun(product_unit,real)),B2),I)) ) ) ) ) ) ).

% compute_powr_real
tff(fact_7303_powr__real__def,axiom,
    powr_real = powr(real) ).

% powr_real_def
tff(fact_7304_gen__length__code_I2_J,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : aa(list(B),nat,gen_length(B,N),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),nat,gen_length(B,aa(nat,nat,suc,N)),Xs) ).

% gen_length_code(2)
tff(fact_7305_length__code,axiom,
    ! [A: $tType] : size_size(list(A)) = gen_length(A,zero_zero(nat)) ).

% length_code
tff(fact_7306_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,X)) = aa(nat,int,semiring_1_of_nat(int),X) ).

% integer_of_nat.rep_eq
tff(fact_7307_int__of__integer__integer__of__nat,axiom,
    ! [N: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,N)) = aa(nat,int,semiring_1_of_nat(int),N) ).

% int_of_integer_integer_of_nat
tff(fact_7308_integer__of__nat__0,axiom,
    aa(nat,code_integer,code_integer_of_nat,zero_zero(nat)) = zero_zero(code_integer) ).

% integer_of_nat_0
tff(fact_7309_integer__of__nat_Oabs__eq,axiom,
    ! [X: nat] : aa(nat,code_integer,code_integer_of_nat,X) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),X)) ).

% integer_of_nat.abs_eq
tff(fact_7310_integer__of__nat__1,axiom,
    aa(nat,code_integer,code_integer_of_nat,one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_7311_integer__of__nat__def,axiom,
    code_integer_of_nat = aa(fun(nat,int),fun(nat,code_integer),map_fun(nat,nat,int,code_integer,id(nat),code_integer_of_int),semiring_1_of_nat(int)) ).

% integer_of_nat_def
tff(fact_7312_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] : collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_ht(nat,fun(nat,fun(nat,bool)),M2))) = product_Sigma(nat,nat,set_ord_atMost(nat,M2),aTP_Lamp_afb(nat,fun(nat,set(nat)),M2)) ).

% pairs_le_eq_Sigma
tff(fact_7313_Sigma__interval__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [A4: set(B),V2: fun(B,A),W: A] : 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))),inf_inf(set(product_prod(B,A))),product_Sigma(B,A,A4,aTP_Lamp_afc(fun(B,A),fun(B,set(A)),V2))),product_Sigma(B,A,A4,aa(A,fun(B,set(A)),aTP_Lamp_afd(fun(B,A),fun(A,fun(B,set(A))),V2),W))) = bot_bot(set(product_prod(B,A))) ) ).

% Sigma_interval_disjoint
tff(fact_7314_product__atMost__eq__Un,axiom,
    ! [A4: set(nat),M2: nat] : product_Sigma(nat,nat,A4,aTP_Lamp_afe(nat,fun(nat,set(nat)),M2)) = aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),sup_sup(set(product_prod(nat,nat))),product_Sigma(nat,nat,A4,aTP_Lamp_afb(nat,fun(nat,set(nat)),M2))),product_Sigma(nat,nat,A4,aTP_Lamp_aff(nat,fun(nat,set(nat)),M2))) ).

% product_atMost_eq_Un
tff(fact_7315_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),I5: set(B)] :
    ? [F3: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),A4,I5)))
      & pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),image(A,product_prod(B,A),F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),A4,I5)))),product_Sigma(B,A,I5,A4))) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_7316_Compl__Times__UNIV1,axiom,
    ! [A: $tType,B: $tType,A4: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_afg(set(B),fun(A,set(B)),A4))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_afh(set(B),fun(A,set(B)),A4)) ).

% Compl_Times_UNIV1
tff(fact_7317_Compl__Times__UNIV2,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_afi(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A4),aTP_Lamp_afi(A,set(B))) ).

% Compl_Times_UNIV2
tff(fact_7318_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Diff_distrib1
tff(fact_7319_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B5: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5),aTP_Lamp_afg(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_afg(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_afg(set(B),fun(A,set(B)),C5))) ).

% Times_Diff_distrib1
tff(fact_7320_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A4: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_afj(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A4),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,A4)),product_Sigma(A,B,I5,B5)) ).

% Sigma_Diff_distrib2
tff(fact_7321_lists__length__Suc__eq,axiom,
    ! [A: $tType,A4: set(A),N: nat] : collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_afk(set(A),fun(nat,fun(list(A),bool)),A4),N)) = image(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_acy(list(A),fun(A,list(A)))),product_Sigma(list(A),A,collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_iy(set(A),fun(nat,fun(list(A),bool)),A4),N)),aTP_Lamp_afl(set(A),fun(list(A),set(A)),A4))) ).

% lists_length_Suc_eq
tff(fact_7322_card__def,axiom,
    ! [B: $tType] : finite_card(B) = finite_folding_F(B,nat,aTP_Lamp_afm(B,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_7323_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F2: fun(A,fun(B,B)),A4: set(A),X: A,Z: B] :
      ( finite_folding_on(A,B,S3,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),S3))
       => ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
           => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A4) = aa(B,B,aa(A,fun(B,B),F2,X),aa(set(A),B,finite_folding_F(A,B,F2,Z),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,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_7324_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F2: fun(A,fun(B,B)),X: A,A4: set(A),Z: B] :
      ( finite_folding_on(A,B,S3,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,X),A4)),S3))
       => ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),insert(A,X),A4)) = aa(B,B,aa(A,fun(B,B),F2,X),aa(set(A),B,finite_folding_F(A,B,F2,Z),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,X),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_7325_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_aea(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_7326_filtermap__ln__at__right,axiom,
    filtermap(real,real,ln_ln(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) = at_bot(real) ).

% filtermap_ln_at_right
tff(fact_7327_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list(int)] : gcd_Gcd(int,aa(list(int),set(int),set2(int),Xs)) = fold(int,int,gcd_gcd(int),Xs,zero_zero(int)) ).

% Gcd_int_set_eq_fold
tff(fact_7328_filtermap__nhds__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A] : filtermap(A,A,uminus_uminus(A),topolo7230453075368039082e_nhds(A,A3)) = topolo7230453075368039082e_nhds(A,aa(A,A,uminus_uminus(A),A3)) ) ).

% filtermap_nhds_minus
tff(fact_7329_filtermap__ln__at__top,axiom,
    filtermap(real,real,ln_ln(real),at_top(real)) = at_top(real) ).

% filtermap_ln_at_top
tff(fact_7330_filtermap__exp__at__top,axiom,
    filtermap(real,real,exp(real),at_top(real)) = at_top(real) ).

% filtermap_exp_at_top
tff(fact_7331_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_7332_filtermap__nhds__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [D2: A,A3: A] : filtermap(A,A,aTP_Lamp_afn(A,fun(A,A),D2),topolo7230453075368039082e_nhds(A,A3)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D2)) ) ).

% filtermap_nhds_shift
tff(fact_7333_at__top__mirror,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add(A)
        & linorder(A) )
     => ( at_top(A) = filtermap(A,A,uminus_uminus(A),at_bot(A)) ) ) ).

% at_top_mirror
tff(fact_7334_at__bot__mirror,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add(A)
        & linorder(A) )
     => ( at_bot(A) = filtermap(A,A,uminus_uminus(A),at_top(A)) ) ) ).

% at_bot_mirror
tff(fact_7335_filtermap__at__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [D2: A,A3: A] : filtermap(A,A,aTP_Lamp_afn(A,fun(A,A),D2),topolo174197925503356063within(A,A3,top_top(set(A)))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D2),top_top(set(A))) ) ).

% filtermap_at_shift
tff(fact_7336_filtermap__at__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A] : filtermap(A,A,uminus_uminus(A),topolo174197925503356063within(A,A3,top_top(set(A)))) = topolo174197925503356063within(A,aa(A,A,uminus_uminus(A),A3),top_top(set(A))) ) ).

% filtermap_at_minus
tff(fact_7337_filtermap__at__right__shift,axiom,
    ! [D2: real,A3: real] : filtermap(real,real,aTP_Lamp_afo(real,fun(real,real),D2),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3))) = topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),D2),set_ord_greaterThan(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),D2))) ).

% filtermap_at_right_shift
tff(fact_7338_at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A] : topolo174197925503356063within(A,A3,top_top(set(A))) = filtermap(A,A,aTP_Lamp_afp(A,fun(A,A),A3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% at_to_0
tff(fact_7339_Gcd__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : gcd_Gcd(A,aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_gcd(A),Xs,zero_zero(A)) ) ).

% Gcd_set_eq_fold
tff(fact_7340_Gcd__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_gcd(A),Xs,zero_zero(A)) ) ).

% Gcd_fin.set_eq_fold
tff(fact_7341_at__right__minus,axiom,
    ! [A3: real] : topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)) = filtermap(real,real,uminus_uminus(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A3),set_ord_lessThan(real,aa(real,real,uminus_uminus(real),A3)))) ).

% at_right_minus
tff(fact_7342_at__left__minus,axiom,
    ! [A3: real] : topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)) = filtermap(real,real,uminus_uminus(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A3),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A3)))) ).

% at_left_minus
tff(fact_7343_filtermap__times__pos__at__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [C2: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7344_at__to__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ( topolo174197925503356063within(A,zero_zero(A),top_top(set(A))) = filtermap(A,A,inverse_inverse(A),at_infinity(A)) ) ) ).

% at_to_infinity
tff(fact_7345_less__than__iff,axiom,
    ! [X: nat,Y2: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Y2)),less_than))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y2)) ) ).

% less_than_iff
tff(fact_7346_elimnum,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),N)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimnum
tff(fact_7347_idiff__enat__0,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_enat2(zero_zero(nat))),N) = extended_enat2(zero_zero(nat)) ).

% idiff_enat_0
tff(fact_7348_idiff__enat__0__right,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),N),extended_enat2(zero_zero(nat))) = N ).

% idiff_enat_0_right
tff(fact_7349_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_7350_enat__ord__simps_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),extended_enat2(M2)),extended_enat2(N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% enat_ord_simps(2)
tff(fact_7351_numeral__less__enat__iff,axiom,
    ! [M2: num,N: nat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M2)),extended_enat2(N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M2)),N)) ) ).

% numeral_less_enat_iff
tff(fact_7352_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_zero(extended_enat) = extended_enat2(X) )
    <=> ( X = zero_zero(nat) ) ) ).

% enat_0_iff(2)
tff(fact_7353_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = zero_zero(extended_enat) )
    <=> ( X = zero_zero(nat) ) ) ).

% enat_0_iff(1)
tff(fact_7354_zero__enat__def,axiom,
    zero_zero(extended_enat) = extended_enat2(zero_zero(nat)) ).

% zero_enat_def
tff(fact_7355_enat__1__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( one_one(extended_enat) = extended_enat2(X) )
    <=> ( X = one_one(nat) ) ) ).

% enat_1_iff(2)
tff(fact_7356_enat__1__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = one_one(extended_enat) )
    <=> ( X = one_one(nat) ) ) ).

% enat_1_iff(1)
tff(fact_7357_one__enat__def,axiom,
    one_one(extended_enat) = extended_enat2(one_one(nat)) ).

% one_enat_def
tff(fact_7358_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A3: bool,B2: bool,Uu: extended_enat] : vEBT_VEBT_elim_dead(vEBT_Leaf(A3,B2),Uu) = vEBT_Leaf(A3,B2) ).

% VEBT_internal.elim_dead.simps(1)
tff(fact_7359_less__enatE,axiom,
    ! [N: extended_enat,M2: nat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),N),extended_enat2(M2)))
     => ~ ! [K2: nat] :
            ( ( N = extended_enat2(K2) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),M2)) ) ) ).

% less_enatE
tff(fact_7360_Suc__ile__eq,axiom,
    ! [M2: nat,N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),extended_enat2(aa(nat,nat,suc,M2))),N))
    <=> pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),extended_enat2(M2)),N)) ) ).

% Suc_ile_eq
tff(fact_7361_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extended_enat2(L)) = vEBT_Node(Info,Deg,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),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_afq(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),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),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_7362_minus__set__fold,axiom,
    ! [A: $tType,A4: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(list(A),set(A),set2(A),Xs)) = fold(A,set(A),remove(A),Xs,A4) ).

% minus_set_fold
tff(fact_7363_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Gcd(nat,aa(list(nat),set(nat),set2(nat),Xs)) = fold(nat,nat,gcd_gcd(nat),Xs,zero_zero(nat)) ).

% Gcd_nat_set_eq_fold
tff(fact_7364_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: extended_enat,Y2: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa) = Y2 )
     => ( ! [A6: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A6,B4) )
           => ( Y2 != vEBT_Leaf(A6,B4) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( ( Xa = extend4730790105801354508finity(extended_enat) )
               => ( Y2 != vEBT_Node(Info2,Deg2,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_afq(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList2),vEBT_VEBT_elim_dead(Summary2,extend4730790105801354508finity(extended_enat))) ) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ! [L2: nat] :
                    ( ( Xa = extended_enat2(L2) )
                   => ( Y2 != vEBT_Node(Info2,Deg2,take(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L2),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),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_afq(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList2)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L2),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),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_7365_elimcomplete,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),N)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extend4730790105801354508finity(extended_enat)) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimcomplete
tff(fact_7366_idiff__infinity,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extend4730790105801354508finity(extended_enat)),N) = extend4730790105801354508finity(extended_enat) ).

% idiff_infinity
tff(fact_7367_add__diff__cancel__enat,axiom,
    ! [X: extended_enat,Y2: extended_enat] :
      ( ( X != 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),X),Y2)),X) = Y2 ) ) ).

% add_diff_cancel_enat
tff(fact_7368_idiff__0,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),zero_zero(extended_enat)),N) = zero_zero(extended_enat) ).

% idiff_0
tff(fact_7369_idiff__0__right,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),N),zero_zero(extended_enat)) = N ).

% idiff_0_right
tff(fact_7370_idiff__self,axiom,
    ! [N: extended_enat] :
      ( ( N != extend4730790105801354508finity(extended_enat) )
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),N),N) = zero_zero(extended_enat) ) ) ).

% idiff_self
tff(fact_7371_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(N)) = zero_zero(extended_enat) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(N)) = extend4730790105801354508finity(extended_enat) ) ) ) ).

% times_enat_simps(3)
tff(fact_7372_times__enat__simps_I4_J,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_enat2(M2)),extend4730790105801354508finity(extended_enat)) = zero_zero(extended_enat) ) )
      & ( ( M2 != zero_zero(nat) )
       => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_enat2(M2)),extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ) ) ) ).

% times_enat_simps(4)
tff(fact_7373_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_7374_infinity__ne__i1,axiom,
    extend4730790105801354508finity(extended_enat) != one_one(extended_enat) ).

% infinity_ne_i1
tff(fact_7375_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_7376_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y2: extended_enat,X: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),Z),Y2))
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),X),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Y2),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),X),Y2)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_7377_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,extended_enat)] :
      ( ! [A6: bool,B4: bool,Uu2: extended_enat] : X != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Leaf(A6,B4)),Uu2)
     => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info2,Deg2,TreeList2,Summary2)),extend4730790105801354508finity(extended_enat))
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,L2: nat] : X != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info2,Deg2,TreeList2,Summary2)),extended_enat2(L2)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_7378_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extend4730790105801354508finity(extended_enat)) = vEBT_Node(Info,Deg,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_afq(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg),TreeList),vEBT_VEBT_elim_dead(Summary,extend4730790105801354508finity(extended_enat))) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_7379_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: extended_enat,Y2: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa) = Y2 )
     => ( accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,X),Xa))
       => ( ! [A6: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A6,B4) )
             => ( ( Y2 = vEBT_Leaf(A6,B4) )
               => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Leaf(A6,B4)),Xa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Xa = extend4730790105801354508finity(extended_enat) )
                 => ( ( Y2 = vEBT_Node(Info2,Deg2,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_afq(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList2),vEBT_VEBT_elim_dead(Summary2,extend4730790105801354508finity(extended_enat))) )
                   => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info2,Deg2,TreeList2,Summary2)),extend4730790105801354508finity(extended_enat))) ) ) )
           => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
                 => ! [L2: nat] :
                      ( ( Xa = extended_enat2(L2) )
                     => ( ( Y2 = vEBT_Node(Info2,Deg2,take(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L2),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),Deg2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_afq(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList2)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),L2),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),Deg2),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),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info2,Deg2,TreeList2,Summary2)),extended_enat2(L2))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_7380_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) = extended_case_enat(extended_enat,aTP_Lamp_afs(extended_enat,fun(nat,extended_enat),B2),extend4730790105801354508finity(extended_enat),A3) ).

% diff_enat_def
tff(fact_7381_times__enat__def,axiom,
    ! [M2: extended_enat,N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M2),N) = extended_case_enat(extended_enat,aTP_Lamp_afu(extended_enat,fun(nat,extended_enat),N),if(extended_enat,aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),fequal(extended_enat),N),zero_zero(extended_enat)),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)),M2) ).

% times_enat_def
tff(fact_7382_eSuc__def,axiom,
    ! [I: extended_enat] : extended_eSuc(I) = extended_case_enat(extended_enat,aTP_Lamp_afv(nat,extended_enat),extend4730790105801354508finity(extended_enat),I) ).

% eSuc_def
tff(fact_7383_binomial__def,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(N),K) = aa(set(set(nat)),nat,finite_card(set(nat)),collect(set(nat),aa(nat,fun(set(nat),bool),aTP_Lamp_afw(nat,fun(nat,fun(set(nat),bool)),N),K))) ).

% binomial_def
tff(fact_7384_eSuc__minus__eSuc,axiom,
    ! [N: extended_enat,M2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_eSuc(N)),extended_eSuc(M2)) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),N),M2) ).

% eSuc_minus_eSuc
tff(fact_7385_eSuc__minus__1,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_eSuc(N)),one_one(extended_enat)) = N ).

% eSuc_minus_1
tff(fact_7386_enat__eSuc__iff,axiom,
    ! [Y2: nat,X: extended_enat] :
      ( ( extended_enat2(Y2) = extended_eSuc(X) )
    <=> ? [N5: nat] :
          ( ( Y2 = aa(nat,nat,suc,N5) )
          & ( extended_enat2(N5) = X ) ) ) ).

% enat_eSuc_iff
tff(fact_7387_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y2: nat] :
      ( ( extended_eSuc(X) = extended_enat2(Y2) )
    <=> ? [N5: nat] :
          ( ( Y2 = aa(nat,nat,suc,N5) )
          & ( X = extended_enat2(N5) ) ) ) ).

% eSuc_enat_iff
tff(fact_7388_eSuc__enat,axiom,
    ! [N: nat] : extended_eSuc(extended_enat2(N)) = extended_enat2(aa(nat,nat,suc,N)) ).

% eSuc_enat
tff(fact_7389_Pow__Compl,axiom,
    ! [A: $tType,A4: set(A)] : pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A4)) = collect(set(A),aTP_Lamp_afx(set(A),fun(set(A),bool),A4)) ).

% Pow_Compl
tff(fact_7390_one__eSuc,axiom,
    one_one(extended_enat) = extended_eSuc(zero_zero(extended_enat)) ).

% one_eSuc
tff(fact_7391_eSuc__plus__1,axiom,
    ! [N: extended_enat] : extended_eSuc(N) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),N),one_one(extended_enat)) ).

% eSuc_plus_1
tff(fact_7392_plus__1__eSuc_I1_J,axiom,
    ! [Q5: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),one_one(extended_enat)),Q5) = extended_eSuc(Q5) ).

% plus_1_eSuc(1)
tff(fact_7393_plus__1__eSuc_I2_J,axiom,
    ! [Q5: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Q5),one_one(extended_enat)) = extended_eSuc(Q5) ).

% plus_1_eSuc(2)
tff(fact_7394_Quotient__real,axiom,
    quotient(fun(nat,rat),real,realrel,real2,rep_real,cr_real) ).

% Quotient_real
tff(fact_7395_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_7396_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),product_Pair(list(A),product_prod(A,list(A)),nil(A)),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),X),Xs))) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(option(product_prod(list(A),product_prod(A,list(A)))),option(product_prod(list(A),product_prod(A,list(A)))),aa(fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(product_prod(list(A),product_prod(A,list(A))))),aa(option(product_prod(list(A),product_prod(A,list(A)))),fun(fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(product_prod(list(A),product_prod(A,list(A)))))),case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A)))),none(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))),aTP_Lamp_afz(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),X))),extract(A,P,Xs)) ) ) ) ).

% extract_Cons_code
tff(fact_7397_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_7398_char__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,char,unique5772411509450598832har_of(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,char,unique5772411509450598832har_of(nat),N) ) ).

% char_of_nat
tff(fact_7399_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_7400_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),N) = char2(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),one_one(nat)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_7401_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_7402_char_Osize_I2_J,axiom,
    ! [X15: bool,X23: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : aa(char,nat,size_size(char),char2(X15,X23,X32,X42,X52,X62,X72,X82)) = zero_zero(nat) ).

% char.size(2)
tff(fact_7403_nat__of__char__less__256,axiom,
    ! [C2: char] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_7404_char_Osize__gen,axiom,
    ! [X15: bool,X23: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : size_char(char2(X15,X23,X32,X42,X52,X62,X72,X82)) = zero_zero(nat) ).

% char.size_gen
tff(fact_7405_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_7406_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_7407_relpow__finite__bounded1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),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_aga(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),collect(nat,aTP_Lamp_agb(set(product_prod(A,A)),fun(nat,bool),R2)))))) ) ) ).

% relpow_finite_bounded1
tff(fact_7408_cr__int__def,axiom,
    ! [X3: product_prod(nat,nat)] : aa(product_prod(nat,nat),fun(int,bool),cr_int,X3) = aa(int,fun(int,bool),fequal(int),aa(product_prod(nat,nat),int,abs_Integ,X3)) ).

% cr_int_def
tff(fact_7409_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_7410_finite__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(set(product_prod(A,A)),bool,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))),N),R2))) ) ) ).

% finite_relpow
tff(fact_7411_relpow__Suc__I2,axiom,
    ! [A: $tType,X: A,Y2: A,R2: set(product_prod(A,A)),Z: A,N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y2),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))),N),R2)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R2))) ) ) ).

% relpow_Suc_I2
tff(fact_7412_relpow__Suc__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R2)))
     => ~ ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y3)),R2))
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% relpow_Suc_E2
tff(fact_7413_relpow__Suc__D2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R2)))
     => ? [Y3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y3)),R2))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% relpow_Suc_D2
tff(fact_7414_relpow__Suc__I,axiom,
    ! [A: $tType,X: A,Y2: A,N: nat,R2: set(product_prod(A,A)),Z: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),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))),N),R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y2),Z)),R2))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R2))) ) ) ).

% relpow_Suc_I
tff(fact_7415_relpow__Suc__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R2)))
     => ~ ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z)),R2)) ) ) ).

% relpow_Suc_E
tff(fact_7416_relpow__0__E,axiom,
    ! [A: $tType,X: A,Y2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y2)),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)))
     => ( X = Y2 ) ) ).

% relpow_0_E
tff(fact_7417_relpow__0__I,axiom,
    ! [A: $tType,X: A,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),X)),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_7418_relpow__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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))),N),R2)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y3: A,M3: nat] :
              ( ( N = aa(nat,nat,suc,M3) )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y3)),R2))
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M3),R2))) ) ) ) ) ).

% relpow_E2
tff(fact_7419_relpow__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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))),N),R2)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y3: A,M3: nat] :
              ( ( N = aa(nat,nat,suc,M3) )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M3),R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z)),R2)) ) ) ) ) ).

% relpow_E
tff(fact_7420_relpow__empty,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( 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))),N),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_7421_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
tff(fact_7422_relpow__fun__conv,axiom,
    ! [A: $tType,A3: A,B2: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),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))),N),R2)))
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = A3 )
          & ( aa(nat,A,F5,N) = B2 )
          & ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
             => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F5,I2)),aa(nat,A,F5,aa(nat,nat,suc,I2)))),R2)) ) ) ) ).

% relpow_fun_conv
tff(fact_7423_ntrancl__def,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,N,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_aga(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),collect(nat,aTP_Lamp_agc(nat,fun(nat,bool),N)))) ).

% ntrancl_def
tff(fact_7424_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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_aga(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),collect(nat,aTP_Lamp_agb(set(product_prod(A,A)),fun(nat,bool),R2)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_7425_ntrancl__Zero,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_ntrancl(A,zero_zero(nat),R2) = R2 ).

% ntrancl_Zero
tff(fact_7426_finite__trancl__ntranl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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_7427_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_7428_trancl__power,axiom,
    ! [A: $tType,P2: product_prod(A,A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P2),transitive_trancl(A,R2)))
    <=> ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),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))),N5),R2))) ) ) ).

% trancl_power
tff(fact_7429_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,M2),N)),transitive_trancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% less_eq
tff(fact_7430_Quotient__int,axiom,
    quotient(product_prod(nat,nat),int,intrel,abs_Integ,rep_Integ,cr_int) ).

% Quotient_int
tff(fact_7431_gfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( 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_7432_gfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),N: nat] :
          ( 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,N)),F2)) = complete_lattice_gfp(A,F2) ) ) ) ).

% gfp_funpow
tff(fact_7433_strict__sorted__equal__Uniq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : uniq(list(A),aTP_Lamp_agd(set(A),fun(list(A),bool),A4)) ) ).

% strict_sorted_equal_Uniq
tff(fact_7434_list__ex__length,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( list_ex(A,P,Xs)
    <=> ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N5))) ) ) ).

% list_ex_length
tff(fact_7435_span__breakdown,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,S3: set(A),A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),S3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_span(A,S3)))
           => ? [K2: real] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),real_V8093663219630862766scaleR(A,K2,B2))),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))))) ) ) ) ).

% span_breakdown
tff(fact_7436_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_7437_span__insert__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] : real_Vector_span(A,aa(set(A),set(A),insert(A,zero_zero(A)),S3)) = real_Vector_span(A,S3) ) ).

% span_insert_0
tff(fact_7438_span__empty,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ( real_Vector_span(A,bot_bot(set(A))) = aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A))) ) ) ).

% span_empty
tff(fact_7439_span__delete__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] : real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A))))) = real_Vector_span(A,S3) ) ).

% span_delete_0
tff(fact_7440_span__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),real_Vector_span(A,S3))) ) ).

% span_0
tff(fact_7441_span__breakdown__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,A3: A,S3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,aa(set(A),set(A),insert(A,A3),S3))))
        <=> ? [K3: real] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),real_V8093663219630862766scaleR(A,K3,A3))),real_Vector_span(A,S3))) ) ) ).

% span_breakdown_eq
tff(fact_7442_in__span__delete,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A,S3: set(A),B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_span(A,S3)))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_span(A,aa(set(A),set(A),insert(A,A3),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,B2),bot_bot(set(A)))))))) ) ) ) ).

% in_span_delete
tff(fact_7443_span__diff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,S3: set(A),Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,S3)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),real_Vector_span(A,S3)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)),real_Vector_span(A,S3))) ) ) ) ).

% span_diff
tff(fact_7444_eq__span__insert__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y2: A,S3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)),real_Vector_span(A,S3)))
         => ( real_Vector_span(A,aa(set(A),set(A),insert(A,X),S3)) = real_Vector_span(A,aa(set(A),set(A),insert(A,Y2),S3)) ) ) ) ).

% eq_span_insert_eq
tff(fact_7445_semilattice__neutr__order_Oaxioms_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => semilattice_neutr(A,F2,Z) ) ).

% semilattice_neutr_order.axioms(1)
tff(fact_7446_span__neg,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,S3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,S3)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,uminus_uminus(A),X)),real_Vector_span(A,S3))) ) ) ).

% span_neg
tff(fact_7447_span__induct__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,S3: set(A),H: fun(A,bool)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,S3)))
         => ( pp(aa(A,bool,H,zero_zero(A)))
           => ( ! [C3: real,X2: A,Y3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S3))
                 => ( pp(aa(A,bool,H,Y3))
                   => pp(aa(A,bool,H,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,C3,X2)),Y3))) ) )
             => pp(aa(A,bool,H,X)) ) ) ) ) ).

% span_induct_alt
tff(fact_7448_inf__top_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => semilattice_neutr(A,inf_inf(A),top_top(A)) ) ).

% inf_top.semilattice_neutr_axioms
tff(fact_7449_sup__bot_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semilattice_neutr(A,sup_sup(A),bot_bot(A)) ) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_7450_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_7451_span__insert,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A,S3: set(A)] : real_Vector_span(A,aa(set(A),set(A),insert(A,A3),S3)) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_age(A,fun(set(A),fun(A,bool)),A3),S3)) ) ).

% span_insert
tff(fact_7452_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_7453_max__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.semilattice_neutr_axioms
tff(fact_7454_dependent__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [P: set(A)] :
          ( real_V358717886546972837endent(A,P)
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),P))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),P),aa(set(A),set(A),insert(A,X4),bot_bot(set(A))))))) ) ) ) ).

% dependent_def
tff(fact_7455_dim__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A)] :
          ( ( ? [B11: set(A)] :
                ( ~ real_V358717886546972837endent(A,B11)
                & ( real_Vector_span(A,B11) = real_Vector_span(A,V) ) )
           => ( real_Vector_dim(A,V) = aa(set(A),nat,finite_card(A),fChoice(set(A),aTP_Lamp_agf(set(A),fun(set(A),bool),V))) ) )
          & ( ~ ? [B4: set(A)] :
                  ( ~ real_V358717886546972837endent(A,B4)
                  & ( real_Vector_span(A,B4) = real_Vector_span(A,V) ) )
           => ( real_Vector_dim(A,V) = zero_zero(nat) ) ) ) ) ).

% dim_def
tff(fact_7456_linear__indep__image__lemma,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),B5: set(A),X: A] :
          ( real_Vector_linear(A,B,F2)
         => ( pp(aa(set(A),bool,finite_finite(A),B5))
           => ( ~ real_V358717886546972837endent(B,image(A,B,F2,B5))
             => ( inj_on(A,B,F2,B5)
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,B5)))
                 => ( ( aa(A,B,F2,X) = zero_zero(B) )
                   => ( X = zero_zero(A) ) ) ) ) ) ) ) ) ).

% linear_indep_image_lemma
tff(fact_7457_linear__eq__0__on__span,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F2: fun(A,B),B2: set(A),X: A] :
          ( real_Vector_linear(A,B,F2)
         => ( ! [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B2))
               => ( aa(A,B,F2,X2) = zero_zero(B) ) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,B2)))
             => ( aa(A,B,F2,X) = zero_zero(B) ) ) ) ) ) ).

% linear_eq_0_on_span
tff(fact_7458_linear__injective__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => ( inj_on(A,B,F2,top_top(set(A)))
          <=> ! [X4: A] :
                ( ( aa(A,B,F2,X4) = zero_zero(B) )
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% linear_injective_0
tff(fact_7459_linear__compose__neg,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => real_Vector_linear(A,B,aTP_Lamp_agg(fun(A,B),fun(A,B),F2)) ) ) ).

% linear_compose_neg
tff(fact_7460_linear__neg,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),X: A] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(A,B,F2,aa(A,A,uminus_uminus(A),X)) = aa(B,B,uminus_uminus(B),aa(A,B,F2,X)) ) ) ) ).

% linear_neg
tff(fact_7461_module__hom__uminus,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => real_Vector_linear(A,A,uminus_uminus(A)) ) ).

% module_hom_uminus
tff(fact_7462_linear__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),X: A,Y2: A] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) ) ) ) ).

% linear_diff
tff(fact_7463_linear__compose__sub,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => ( real_Vector_linear(A,B,G)
           => real_Vector_linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agh(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% linear_compose_sub
tff(fact_7464_linear__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(A,B,F2,zero_zero(A)) = zero_zero(B) ) ) ) ).

% linear_0
tff(fact_7465_module__hom__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => real_Vector_linear(A,B,aTP_Lamp_agi(A,B)) ) ).

% module_hom_zero
tff(fact_7466_representation__diff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A,U: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),real_Vector_span(A,Basis)))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),real_Vector_span(A,Basis)))
             => ! [X3: A] : real_V7696804695334737415tation(A,Basis,aa(A,A,aa(A,fun(A,A),minus_minus(A),U),V2),X3) = aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7696804695334737415tation(A,Basis,U,X3)),real_V7696804695334737415tation(A,Basis,V2,X3)) ) ) ) ) ).

% representation_diff
tff(fact_7467_representation__neg,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),real_Vector_span(A,Basis)))
           => ! [X3: A] : real_V7696804695334737415tation(A,Basis,aa(A,A,uminus_uminus(A),V2),X3) = aa(real,real,uminus_uminus(real),real_V7696804695334737415tation(A,Basis,V2,X3)) ) ) ) ).

% representation_neg
tff(fact_7468_representation__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),X3: A] : real_V7696804695334737415tation(A,Basis,zero_zero(A),X3) = zero_zero(real) ) ).

% representation_zero
tff(fact_7469_representation__basis,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),B2: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),Basis))
           => ! [X3: A] :
                ( ( ( X3 = B2 )
                 => ( real_V7696804695334737415tation(A,Basis,B2,X3) = one_one(real) ) )
                & ( ( X3 != B2 )
                 => ( real_V7696804695334737415tation(A,Basis,B2,X3) = zero_zero(real) ) ) ) ) ) ) ).

% representation_basis
tff(fact_7470_finite__sequence__to__countable__set,axiom,
    ! [A: $tType,X6: set(A)] :
      ( countable_countable(A,X6)
     => ~ ! [F8: fun(nat,set(A))] :
            ( ! [I4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F8,I4)),X6))
           => ( ! [I4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F8,I4)),aa(nat,set(A),F8,aa(nat,nat,suc,I4))))
             => ( ! [I4: nat] : pp(aa(set(A),bool,finite_finite(A),aa(nat,set(A),F8,I4)))
               => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),F8,top_top(set(nat)))) != X6 ) ) ) ) ) ).

% finite_sequence_to_countable_set
tff(fact_7471_construct__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [B5: set(A),G: fun(A,B),V2: A] : real_V4425403222259421789struct(A,B,B5,G,V2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_agj(set(A),fun(fun(A,B),fun(A,fun(A,B))),B5),G),V2)),collect(A,aa(A,fun(A,bool),aTP_Lamp_agk(set(A),fun(A,fun(A,bool)),B5),V2))) ) ).

% construct_def
tff(fact_7472_countable__Diff,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( countable_countable(A,A4)
     => countable_countable(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) ) ).

% countable_Diff
tff(fact_7473_countable__Diff__eq,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( countable_countable(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,X),bot_bot(set(A)))))
    <=> countable_countable(A,A4) ) ).

% countable_Diff_eq
tff(fact_7474_less__ccSup__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S3: set(A),A3: A] :
          ( countable_countable(A,S3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),S3)))
          <=> ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X4)) ) ) ) ) ).

% less_ccSup_iff
tff(fact_7475_ccInf__less__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S3: set(A),A3: A] :
          ( countable_countable(A,S3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S3)),A3))
          <=> ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A3)) ) ) ) ) ).

% ccInf_less_iff
tff(fact_7476_uncountable__minus__countable,axiom,
    ! [A: $tType,A4: set(A),B5: set(A)] :
      ( ~ countable_countable(A,A4)
     => ( countable_countable(A,B5)
       => ~ countable_countable(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) ) ) ).

% uncountable_minus_countable
tff(fact_7477_less__ccSUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [A4: set(B),A3: A,F2: fun(B,A)] :
          ( countable_countable(B,A4)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))))
          <=> ? [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F2,X4))) ) ) ) ) ).

% less_ccSUP_iff
tff(fact_7478_ccINF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [A4: set(B),F2: fun(B,A),A3: A] :
          ( countable_countable(B,A4)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))),A3))
          <=> ? [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X4)),A3)) ) ) ) ) ).

% ccINF_less_iff
tff(fact_7479_construct__outside,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [B5: set(A),V2: A,F2: fun(A,B)] :
          ( ~ real_V358717886546972837endent(A,B5)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),real_V4986007116245087402_basis(A,B5)),B5))))
           => ( real_V4425403222259421789struct(A,B,B5,F2,V2) = zero_zero(B) ) ) ) ) ).

% construct_outside
tff(fact_7480_uminus__integer__def,axiom,
    uminus_uminus(code_integer) = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),uminus_uminus(int)) ).

% uminus_integer_def
tff(fact_7481_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,ord_less(nat))) ).

% natLess_def
tff(fact_7482_Restr__natLeq,axiom,
    ! [N: 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,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N)),aTP_Lamp_agl(nat,fun(nat,set(nat)),N))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_agm(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq
tff(fact_7483_divide__integer__def,axiom,
    divide_divide(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),divide_divide(int)) ).

% divide_integer_def
tff(fact_7484_minus__integer__def,axiom,
    minus_minus(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),minus_minus(int)) ).

% minus_integer_def
tff(fact_7485_Restr__natLeq2,axiom,
    ! [N: 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,N),aTP_Lamp_agn(nat,fun(nat,set(nat)),N))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_agm(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq2
tff(fact_7486_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F2: fun(fun(A,B),C),G: C] :
          ( ! [X2: fun(A,B)] : aa(fun(A,B),C,F2,X2) = G
         => ( aa(fun(A,B),C,F2,aTP_Lamp_ago(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_7487_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,N) = collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N)) ).

% natLeq_underS_less
tff(fact_7488_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_7489_sub_Oabs__eq,axiom,
    ! [Xa: num,X: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Xa),X) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Xa)),aa(num,int,numeral_numeral(int),X))) ).

% sub.abs_eq
tff(fact_7490_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_7491_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_7492_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_7493_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_7494_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_7495_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_7496_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_7497_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_7498_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_7499_sub_Orep__eq,axiom,
    ! [X: num,Xa: num] : aa(code_integer,int,code_int_of_integer,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,X),Xa)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),X)),aa(num,int,numeral_numeral(int),Xa)) ).

% sub.rep_eq
tff(fact_7500_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M2: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit0,M2)),aa(num,num,bit1,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),code_dup(aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M2),N))),one_one(code_integer)) ).

% Code_Numeral.sub_code(9)
tff(fact_7501_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M2: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit1,M2)),aa(num,num,bit0,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_dup(aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M2),N))),one_one(code_integer)) ).

% Code_Numeral.sub_code(8)
tff(fact_7502_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_7503_inv__o__cancel,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(fun(A,B),fun(A,A),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F2)),F2) = id(A) ) ) ).

% inv_o_cancel
tff(fact_7504_inv__into__f__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),X: A] :
      ( inj_on(A,B,F2,A4)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
       => ( aa(B,A,hilbert_inv_into(A,B,A4,F2),aa(A,B,F2,X)) = X ) ) ) ).

% inv_into_f_f
tff(fact_7505_inv__identity,axiom,
    ! [A: $tType,X3: A] : aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),aTP_Lamp_np(A,A)),X3) = X3 ).

% inv_identity
tff(fact_7506_inv__id,axiom,
    ! [A: $tType] : hilbert_inv_into(A,A,top_top(set(A)),id(A)) = id(A) ).

% inv_id
tff(fact_7507_inv__into__image__cancel,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),S3: set(A)] :
      ( inj_on(A,B,F2,A4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S3),A4))
       => ( image(B,A,hilbert_inv_into(A,B,A4,F2),image(A,B,F2,S3)) = S3 ) ) ) ).

% inv_into_image_cancel
tff(fact_7508_o__inv__o__cancel,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(A,B),G: fun(A,C)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(fun(A,B),fun(A,C),comp(B,C,A,aa(fun(B,A),fun(B,C),comp(A,C,B,G),hilbert_inv_into(A,B,top_top(set(A)),F2))),F2) = G ) ) ).

% o_inv_o_cancel
tff(fact_7509_inv__fn,axiom,
    ! [A: $tType,F2: fun(A,A),N: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => ( hilbert_inv_into(A,A,top_top(set(A)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F2)) ) ) ).

% inv_fn
tff(fact_7510_bij__betw__inv__into__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),A13: set(B),B5: set(A),B12: set(B)] :
      ( bij_betw(A,B,F2,A4,A13)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A4))
       => ( ( image(A,B,F2,B5) = B12 )
         => bij_betw(B,A,hilbert_inv_into(A,B,A4,F2),B12,B5) ) ) ) ).

% bij_betw_inv_into_subset
tff(fact_7511_inj__on__inv__into,axiom,
    ! [B: $tType,A: $tType,B5: set(A),F2: fun(B,A),A4: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),image(B,A,F2,A4)))
     => inj_on(A,B,hilbert_inv_into(B,A,A4,F2),B5) ) ).

% inj_on_inv_into
tff(fact_7512_image__inv__into__cancel,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B),A13: set(A),B12: set(A)] :
      ( ( image(B,A,F2,A4) = A13 )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B12),A13))
       => ( image(B,A,F2,image(A,B,hilbert_inv_into(B,A,A4,F2),B12)) = B12 ) ) ) ).

% image_inv_into_cancel
tff(fact_7513_inj__transfer,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),P: fun(A,bool),X: A] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ! [Y3: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),image(A,B,F2,top_top(set(A)))))
           => pp(aa(A,bool,P,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),Y3))) )
       => pp(aa(A,bool,P,X)) ) ) ).

% inj_transfer
tff(fact_7514_image__inv__f__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( image(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),image(A,B,F2,A4)) = A4 ) ) ).

% image_inv_f_f
tff(fact_7515_inj__imp__surj__inv,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( image(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),top_top(set(B))) = top_top(set(A)) ) ) ).

% inj_imp_surj_inv
tff(fact_7516_surj__imp__inj__inv,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => inj_on(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),top_top(set(A))) ) ).

% surj_imp_inj_inv
tff(fact_7517_surj__imp__inv__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),G: fun(A,B)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => ( ! [X2: B] : aa(A,B,G,aa(B,A,F2,X2)) = X2
       => ( hilbert_inv_into(B,A,top_top(set(B)),F2) = G ) ) ) ).

% surj_imp_inv_eq
tff(fact_7518_image__f__inv__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(A)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => ( image(B,A,F2,image(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),A4)) = A4 ) ) ).

% image_f_inv_f
tff(fact_7519_surj__iff__all,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
    <=> ! [X4: A] : aa(B,A,F2,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),X4)) = X4 ) ).

% surj_iff_all
tff(fact_7520_surj__f__inv__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Y2: A] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => ( aa(B,A,F2,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),Y2)) = Y2 ) ) ).

% surj_f_inv_f
tff(fact_7521_inv__into__injective,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(A,B),X: B,Y2: B] :
      ( ( aa(B,A,hilbert_inv_into(A,B,A4,F2),X) = aa(B,A,hilbert_inv_into(A,B,A4,F2),Y2) )
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),image(A,B,F2,A4)))
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y2),image(A,B,F2,A4)))
         => ( X = Y2 ) ) ) ) ).

% inv_into_injective
tff(fact_7522_inv__into__into,axiom,
    ! [A: $tType,B: $tType,X: A,F2: fun(B,A),A4: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),image(B,A,F2,A4)))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,hilbert_inv_into(B,A,A4,F2),X)),A4)) ) ).

% inv_into_into
tff(fact_7523_f__inv__into__f,axiom,
    ! [B: $tType,A: $tType,Y2: A,F2: fun(B,A),A4: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),image(B,A,F2,A4)))
     => ( aa(B,A,F2,aa(A,B,hilbert_inv_into(B,A,A4,F2),Y2)) = Y2 ) ) ).

% f_inv_into_f
tff(fact_7524_inv__into__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(A,B),G: fun(C,A),A4: set(C),X: B] :
      ( inj_on(A,B,F2,image(C,A,G,A4))
     => ( inj_on(C,A,G,A4)
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),image(A,B,F2,image(C,A,G,A4))))
         => ( aa(B,C,hilbert_inv_into(C,B,A4,aa(fun(C,A),fun(C,B),comp(A,B,C,F2),G)),X) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,hilbert_inv_into(C,A,A4,G)),hilbert_inv_into(A,B,image(C,A,G,A4),F2)),X) ) ) ) ) ).

% inv_into_comp
tff(fact_7525_inj__imp__inv__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),G: fun(B,A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ! [X2: B] : aa(A,B,F2,aa(B,A,G,X2)) = X2
       => ( hilbert_inv_into(A,B,top_top(set(A)),F2) = G ) ) ) ).

% inj_imp_inv_eq
tff(fact_7526_inv__f__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X: A,Y2: B] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(A,B,F2,X) = Y2 )
       => ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),Y2) = X ) ) ) ).

% inv_f_eq
tff(fact_7527_inv__f__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X: A] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),aa(A,B,F2,X)) = X ) ) ).

% inv_f_f
tff(fact_7528_inv__into__f__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),X: A,Y2: B] :
      ( inj_on(A,B,F2,A4)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
       => ( ( aa(A,B,F2,X) = Y2 )
         => ( aa(B,A,hilbert_inv_into(A,B,A4,F2),Y2) = X ) ) ) ) ).

% inv_into_f_eq
tff(fact_7529_mult_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => monoid(A,times_times(A),one_one(A)) ) ).

% mult.monoid_axioms
tff(fact_7530_add_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.monoid_axioms
tff(fact_7531_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_7532_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_7533_bij__betw__inv__into,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B5: set(B)] :
      ( bij_betw(A,B,F2,A4,B5)
     => bij_betw(B,A,hilbert_inv_into(A,B,A4,F2),B5,A4) ) ).

% bij_betw_inv_into
tff(fact_7534_inv__into__inv__into__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),A13: set(B),A3: A] :
      ( bij_betw(A,B,F2,A4,A13)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
       => ( aa(A,B,hilbert_inv_into(B,A,A13,hilbert_inv_into(A,B,A4,F2)),A3) = aa(A,B,F2,A3) ) ) ) ).

% inv_into_inv_into_eq
tff(fact_7535_bij__betw__inv__into__left,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),A13: set(B),A3: A] :
      ( bij_betw(A,B,F2,A4,A13)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
       => ( aa(B,A,hilbert_inv_into(A,B,A4,F2),aa(A,B,F2,A3)) = A3 ) ) ) ).

% bij_betw_inv_into_left
tff(fact_7536_bij__betw__inv__into__right,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),A13: set(B),A7: B] :
      ( bij_betw(A,B,F2,A4,A13)
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A7),A13))
       => ( aa(A,B,F2,aa(B,A,hilbert_inv_into(A,B,A4,F2),A7)) = A7 ) ) ) ).

% bij_betw_inv_into_right
tff(fact_7537_sup__bot_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => monoid(A,sup_sup(A),bot_bot(A)) ) ).

% sup_bot.monoid_axioms
tff(fact_7538_bij__imp__bij__inv,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),top_top(set(B)),top_top(set(A))) ) ).

% bij_imp_bij_inv
tff(fact_7539_bij__inv__eq__iff,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,B),X: A,Y2: B] :
      ( bij_betw(A,B,P2,top_top(set(A)),top_top(set(B)))
     => ( ( X = aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),P2),Y2) )
      <=> ( aa(A,B,P2,X) = Y2 ) ) ) ).

% bij_inv_eq_iff
tff(fact_7540_inv__inv__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( hilbert_inv_into(B,A,top_top(set(B)),hilbert_inv_into(A,B,top_top(set(A)),F2)) = F2 ) ) ).

% inv_inv_eq
tff(fact_7541_mono__inv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B)] :
          ( order_mono(A,B,F2)
         => ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
           => order_mono(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2)) ) ) ) ).

% mono_inv
tff(fact_7542_o__inv__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G: fun(C,A)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( bij_betw(C,A,G,top_top(set(C)),top_top(set(A)))
       => ( hilbert_inv_into(C,B,top_top(set(C)),aa(fun(C,A),fun(C,B),comp(A,B,C,F2),G)) = aa(fun(B,A),fun(B,C),comp(A,C,B,hilbert_inv_into(C,A,top_top(set(C)),G)),hilbert_inv_into(A,B,top_top(set(A)),F2)) ) ) ) ).

% o_inv_distrib
tff(fact_7543_inv__equality,axiom,
    ! [A: $tType,B: $tType,G: fun(B,A),F2: fun(A,B)] :
      ( ! [X2: A] : aa(B,A,G,aa(A,B,F2,X2)) = X2
     => ( ! [Y3: B] : aa(A,B,F2,aa(B,A,G,Y3)) = Y3
       => ( hilbert_inv_into(A,B,top_top(set(A)),F2) = G ) ) ) ).

% inv_equality
tff(fact_7544_inv__unique__comp,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),G: fun(A,B)] :
      ( ( aa(fun(A,B),fun(A,A),comp(B,A,A,F2),G) = id(A) )
     => ( ( aa(fun(B,A),fun(B,B),comp(A,B,B,G),F2) = id(B) )
       => ( hilbert_inv_into(B,A,top_top(set(B)),F2) = G ) ) ) ).

% inv_unique_comp
tff(fact_7545_gcd__nat_Omonoid__axioms,axiom,
    monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.monoid_axioms
tff(fact_7546_inv__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),X3: A] : aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),X3) = fChoice(B,aa(A,fun(B,bool),aTP_Lamp_agp(fun(B,A),fun(A,fun(B,bool)),F2),X3)) ).

% inv_def
tff(fact_7547_inv__into__def,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(A,B),X3: B] : aa(B,A,hilbert_inv_into(A,B,A4,F2),X3) = fChoice(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_mx(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A4),F2),X3)) ).

% inv_into_def
tff(fact_7548_inv__into__def2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(A,B),X: B] : aa(B,A,hilbert_inv_into(A,B,A4,F2),X) = fChoice(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_mx(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A4),F2),X)) ).

% inv_into_def2
tff(fact_7549_or_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.monoid_axioms
tff(fact_7550_inf__top_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => monoid(A,inf_inf(A),top_top(A)) ) ).

% inf_top.monoid_axioms
tff(fact_7551_xor_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.monoid_axioms
tff(fact_7552_bij__image__Collect__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),P: fun(A,bool)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( image(A,B,F2,collect(A,P)) = collect(B,aa(fun(A,bool),fun(B,bool),aTP_Lamp_agq(fun(A,B),fun(fun(A,bool),fun(B,bool)),F2),P)) ) ) ).

% bij_image_Collect_eq
tff(fact_7553_max__nat_Omonoid__axioms,axiom,
    monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.monoid_axioms
tff(fact_7554_surj__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
    <=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,F2),hilbert_inv_into(B,A,top_top(set(B)),F2)) = id(A) ) ) ).

% surj_iff
tff(fact_7555_inj__imp__bij__betw__inv,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),M10: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),image(A,B,F2,M10),M10) ) ).

% inj_imp_bij_betw_inv
tff(fact_7556_inj__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
    <=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F2)),F2) = id(A) ) ) ).

% inj_iff
tff(fact_7557_bij__vimage__eq__inv__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( vimage(A,B,F2,A4) = image(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),A4) ) ) ).

% bij_vimage_eq_inv_image
tff(fact_7558_inv__fn__o__fn__is__id,axiom,
    ! [A: $tType,F2: fun(A,A),N: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => ! [X3: A] : aa(A,A,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)),N),hilbert_inv_into(A,A,top_top(set(A)),F2))),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)),X3) = X3 ) ).

% inv_fn_o_fn_is_id
tff(fact_7559_fn__o__inv__fn__is__id,axiom,
    ! [A: $tType,F2: fun(A,A),N: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => ! [X3: A] : aa(A,A,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)),N),F2)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F2))),X3) = X3 ) ).

% fn_o_inv_fn_is_id
tff(fact_7560_strict__mono__inv__on__range,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
         => strict_mono_on(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),image(A,B,F2,top_top(set(A)))) ) ) ).

% strict_mono_inv_on_range
tff(fact_7561_bijection_Oinv__comp__right,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,F2),hilbert_inv_into(A,A,top_top(set(A)),F2)) = id(A) ) ) ).

% bijection.inv_comp_right
tff(fact_7562_bijection_Oinv__comp__left,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,hilbert_inv_into(A,A,top_top(set(A)),F2)),F2) = id(A) ) ) ).

% bijection.inv_comp_left
tff(fact_7563_bijection_Oeq__invI,axiom,
    ! [A: $tType,F2: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A3) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),B2) )
       => ( A3 = B2 ) ) ) ).

% bijection.eq_invI
tff(fact_7564_bijection_Oinv__left,axiom,
    ! [A: $tType,F2: fun(A,A),A3: A] :
      ( hilbert_bijection(A,F2)
     => ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),aa(A,A,F2,A3)) = A3 ) ) ).

% bijection.inv_left
tff(fact_7565_bijection_Oinv__right,axiom,
    ! [A: $tType,F2: fun(A,A),A3: A] :
      ( hilbert_bijection(A,F2)
     => ( aa(A,A,F2,aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A3)) = A3 ) ) ).

% bijection.inv_right
tff(fact_7566_bijection_Oeq__inv__iff,axiom,
    ! [A: $tType,F2: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A3) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),B2) )
      <=> ( A3 = B2 ) ) ) ).

% bijection.eq_inv_iff
tff(fact_7567_bijection_Oinv__left__eq__iff,axiom,
    ! [A: $tType,F2: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A3) = B2 )
      <=> ( aa(A,A,F2,B2) = A3 ) ) ) ).

% bijection.inv_left_eq_iff
tff(fact_7568_bijection_Oinv__right__eq__iff,axiom,
    ! [A: $tType,F2: fun(A,A),B2: A,A3: A] :
      ( hilbert_bijection(A,F2)
     => ( ( B2 = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A3) )
      <=> ( aa(A,A,F2,B2) = A3 ) ) ) ).

% bijection.inv_right_eq_iff
tff(fact_7569_bijection__def,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
    <=> bij_betw(A,A,F2,top_top(set(A)),top_top(set(A))) ) ).

% bijection_def
tff(fact_7570_bijection_Ointro,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => hilbert_bijection(A,F2) ) ).

% bijection.intro
tff(fact_7571_bijection_Obij,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => bij_betw(A,A,F2,top_top(set(A)),top_top(set(A))) ) ).

% bijection.bij
tff(fact_7572_bijection_OeqI,axiom,
    ! [A: $tType,F2: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,F2,A3) = aa(A,A,F2,B2) )
       => ( A3 = B2 ) ) ) ).

% bijection.eqI
tff(fact_7573_bijection_Oeq__iff,axiom,
    ! [A: $tType,F2: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,F2,A3) = aa(A,A,F2,B2) )
      <=> ( A3 = B2 ) ) ) ).

% bijection.eq_iff
tff(fact_7574_bijection_Oinj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => inj_on(A,A,F2,top_top(set(A))) ) ).

% bijection.inj
tff(fact_7575_bijection_Osurj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => ( image(A,A,F2,top_top(set(A))) = top_top(set(A)) ) ) ).

% bijection.surj
tff(fact_7576_bijection_Osurj__inv,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => ( image(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),top_top(set(A))) = top_top(set(A)) ) ) ).

% bijection.surj_inv
tff(fact_7577_bijection_Oinj__inv,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => inj_on(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),top_top(set(A))) ) ).

% bijection.inj_inv
tff(fact_7578_bijection_Obij__inv,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => bij_betw(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),top_top(set(A)),top_top(set(A))) ) ).

% bijection.bij_inv
tff(fact_7579_convergent__realpow,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => topolo6863149650580417670ergent(real,aa(real,fun(nat,real),power_power(real),X)) ) ) ).

% convergent_realpow
tff(fact_7580_less__cSUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A4: set(B),F2: fun(B,A),A3: A] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,image(B,A,F2,A4))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))))
            <=> ? [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F2,X4))) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_7581_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X6)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(set(A),A,complete_Sup_Sup(A),X6)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X4)) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_7582_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_uy(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_iff
tff(fact_7583_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_uz(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_right_iff
tff(fact_7584_convergent__minus__iff,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [X6: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X6)
        <=> topolo6863149650580417670ergent(A,aTP_Lamp_agr(fun(nat,A),fun(nat,A),X6)) ) ) ).

% convergent_minus_iff
tff(fact_7585_convergent__Suc__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,aTP_Lamp_vb(fun(nat,A),fun(nat,A),F2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_Suc_iff
tff(fact_7586_convergent__diff__const__right__iff,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( topolo6863149650580417670ergent(A,aa(A,fun(nat,A),aTP_Lamp_ags(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_diff_const_right_iff
tff(fact_7587_convergent__diff,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [X6: fun(nat,A),Y5: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X6)
         => ( topolo6863149650580417670ergent(A,Y5)
           => topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_agt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),X6),Y5)) ) ) ) ).

% convergent_diff
tff(fact_7588_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Y2: A,I: B] :
          ( condit941137186595557371_above(A,image(B,A,F2,A4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A4))),Y2))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,I)),Y2)) ) ) ) ) ).

% cSUP_lessD
tff(fact_7589_map__conv__bind__option,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),X) = aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),X),aa(fun(B,A),fun(B,option(A)),comp(A,option(A),B,some(A)),F2)) ).

% map_conv_bind_option
tff(fact_7590_cINF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A4: set(B),F2: fun(B,A),A3: A] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,image(B,A,F2,A4))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))),A3))
            <=> ? [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X4)),A3)) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_7591_bind__assoc,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: 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),X),F2)),G) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),X),aa(fun(B,option(A)),fun(C,option(A)),aTP_Lamp_agu(fun(C,option(B)),fun(fun(B,option(A)),fun(C,option(A))),F2),G)) ).

% bind_assoc
tff(fact_7592_bind__runit,axiom,
    ! [A: $tType,X: option(A)] : aa(fun(A,option(A)),option(A),aa(option(A),fun(fun(A,option(A)),option(A)),bind(A,A),X),some(A)) = X ).

% bind_runit
tff(fact_7593_bind__rzero,axiom,
    ! [B: $tType,A: $tType,X: option(B)] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),X),aTP_Lamp_oj(B,option(A))) = none(A) ).

% bind_rzero
tff(fact_7594_bdd__above__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X6: set(A)] :
          ( condit941137186595557371_above(A,image(A,A,uminus_uminus(A),X6))
        <=> condit1013018076250108175_below(A,X6) ) ) ).

% bdd_above_uminus
tff(fact_7595_bdd__below__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X6: set(A)] :
          ( condit1013018076250108175_below(A,image(A,A,uminus_uminus(A),X6))
        <=> condit941137186595557371_above(A,X6) ) ) ).

% bdd_below_uminus
tff(fact_7596_bind__map__option,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,B),X: 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),X)),G) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),X),aa(fun(C,B),fun(C,option(A)),comp(B,option(A),C,G),F2)) ).

% bind_map_option
tff(fact_7597_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X6)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y2))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2)) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_7598_bind__split__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(option(A),bool),M2: option(B),F2: fun(B,option(A))] :
      ( pp(aa(option(A),bool,P,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),M2),F2)))
    <=> ~ ( ( ( M2 = none(B) )
            & ~ pp(aa(option(A),bool,P,none(A))) )
          | ? [X4: B] :
              ( ( M2 = aa(B,option(B),some(B),X4) )
              & ~ pp(aa(option(A),bool,P,aa(B,option(A),F2,X4))) ) ) ) ).

% bind_split_asm
tff(fact_7599_bind__split,axiom,
    ! [A: $tType,B: $tType,P: fun(option(A),bool),M2: option(B),F2: fun(B,option(A))] :
      ( pp(aa(option(A),bool,P,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),M2),F2)))
    <=> ( ( ( M2 = none(B) )
         => pp(aa(option(A),bool,P,none(A))) )
        & ! [V5: B] :
            ( ( M2 = aa(B,option(B),some(B),V5) )
           => pp(aa(option(A),bool,P,aa(B,option(A),F2,V5))) ) ) ) ).

% bind_split
tff(fact_7600_bind__eq__Some__conv,axiom,
    ! [B: $tType,A: $tType,F2: option(B),G: fun(B,option(A)),X: 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),X) )
    <=> ? [Y4: B] :
          ( ( F2 = aa(B,option(B),some(B),Y4) )
          & ( aa(B,option(A),G,Y4) = aa(A,option(A),some(A),X) ) ) ) ).

% bind_eq_Some_conv
tff(fact_7601_Option_Obind__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y2: option(A),F2: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( X = Y2 )
     => ( ! [A6: A] :
            ( ( Y2 = aa(A,option(A),some(A),A6) )
           => ( aa(A,option(B),F2,A6) = aa(A,option(B),G,A6) ) )
       => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),X),F2) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y2),G) ) ) ) ).

% Option.bind_cong
tff(fact_7602_bind_Obind__lunit,axiom,
    ! [B: $tType,A: $tType,X: A,F2: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),aa(A,option(A),some(A),X)),F2) = aa(A,option(B),F2,X) ).

% bind.bind_lunit
tff(fact_7603_bind_Obind__lzero,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),none(A)),F2) = none(B) ).

% bind.bind_lzero
tff(fact_7604_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_7605_bind__option__cong__code,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y2: option(A),F2: fun(A,option(B))] :
      ( ( X = Y2 )
     => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),X),F2) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y2),F2) ) ) ).

% bind_option_cong_code
tff(fact_7606_map__option__bind,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(B,A),X: 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),X),G)) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),X),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_7607_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Y2: A,I: B] :
          ( condit1013018076250108175_below(A,image(B,A,F2,A4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A4))))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(B,A,F2,I))) ) ) ) ) ).

% less_cINF_D
tff(fact_7608_set__bind__option,axiom,
    ! [A: $tType,B: $tType,X: 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),X),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),X))) ).

% set_bind_option
tff(fact_7609_combine__options__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),X: option(A),Y2: option(A)] : combine_options(A,F2,X,Y2) = 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),Y2),aa(option(A),fun(A,option(A)),aTP_Lamp_agw(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),F2),Y2)),X) ).

% combine_options_def
tff(fact_7610_elem__set,axiom,
    ! [A: $tType,X: A,Xo: option(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(option(A),set(A),set_option(A),Xo)))
    <=> ( Xo = aa(A,option(A),some(A),X) ) ) ).

% elem_set
tff(fact_7611_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_7612_combine__options__simps_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),X: option(A)] : combine_options(A,F2,X,none(A)) = X ).

% combine_options_simps(2)
tff(fact_7613_combine__options__simps_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Y2: option(A)] : combine_options(A,F2,none(A),Y2) = Y2 ).

% combine_options_simps(1)
tff(fact_7614_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_7615_bind__option__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y2: option(A),F2: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( X = Y2 )
     => ( ! [Z3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(option(A),set(A),set_option(A),Y2)))
           => ( 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),X),F2) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y2),G) ) ) ) ).

% bind_option_cong
tff(fact_7616_option_Oset__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),V2: option(A)] : aa(option(B),set(B),set_option(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),V2)) = image(A,B,F2,aa(option(A),set(A),set_option(A),V2)) ).

% option.set_map
tff(fact_7617_map__option__idI,axiom,
    ! [A: $tType,X: option(A),F2: fun(A,A)] :
      ( ! [Y3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),aa(option(A),set(A),set_option(A),X)))
         => ( aa(A,A,F2,Y3) = Y3 ) )
     => ( aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),F2),X) = X ) ) ).

% map_option_idI
tff(fact_7618_option_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Xa: option(A),F2: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z3: A,Za: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(option(A),set(A),set_option(A),X)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Za),aa(option(A),set(A),set_option(A),Xa)))
           => ( ( aa(A,B,F2,Z3) = aa(A,B,Fa,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),Fa),Xa) )
       => ( X = Xa ) ) ) ).

% option.inj_map_strong
tff(fact_7619_option_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,X: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [Z3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(option(A),set(A),set_option(A),X)))
         => ( 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),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),X) ) ) ).

% option.map_cong0
tff(fact_7620_option_Omap__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Ya: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( ! [Z3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),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),X) = 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_7621_combine__options__left__commute,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Y2: option(A),X: option(A),Z: option(A)] :
      ( ! [X2: A,Y3: A] : aa(A,A,aa(A,fun(A,A),F2,X2),Y3) = aa(A,A,aa(A,fun(A,A),F2,Y3),X2)
     => ( ! [X2: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,X2),Y3)),Z3) = aa(A,A,aa(A,fun(A,A),F2,X2),aa(A,A,aa(A,fun(A,A),F2,Y3),Z3))
       => ( combine_options(A,F2,Y2,combine_options(A,F2,X,Z)) = combine_options(A,F2,X,combine_options(A,F2,Y2,Z)) ) ) ) ).

% combine_options_left_commute
tff(fact_7622_combine__options__commute,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),X: option(A),Y2: option(A)] :
      ( ! [X2: A,Y3: A] : aa(A,A,aa(A,fun(A,A),F2,X2),Y3) = aa(A,A,aa(A,fun(A,A),F2,Y3),X2)
     => ( combine_options(A,F2,X,Y2) = combine_options(A,F2,Y2,X) ) ) ).

% combine_options_commute
tff(fact_7623_combine__options__assoc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),X: option(A),Y2: option(A),Z: option(A)] :
      ( ! [X2: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,X2),Y3)),Z3) = aa(A,A,aa(A,fun(A,A),F2,X2),aa(A,A,aa(A,fun(A,A),F2,Y3),Z3))
     => ( combine_options(A,F2,combine_options(A,F2,X,Y2),Z) = combine_options(A,F2,X,combine_options(A,F2,Y2,Z)) ) ) ).

% combine_options_assoc
tff(fact_7624_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_7625_option_Oset__cases,axiom,
    ! [A: $tType,E2: A,A3: option(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),E2),aa(option(A),set(A),set_option(A),A3)))
     => ( A3 = aa(A,option(A),some(A),E2) ) ) ).

% option.set_cases
tff(fact_7626_option_Oset__intros,axiom,
    ! [A: $tType,X23: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X23),aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X23)))) ).

% option.set_intros
tff(fact_7627_ospec,axiom,
    ! [A: $tType,A4: option(A),P: fun(A,bool),X: A] :
      ( ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(option(A),set(A),set_option(A),A4)))
         => pp(aa(A,bool,P,X2)) )
     => ( ( A4 = aa(A,option(A),some(A),X) )
       => pp(aa(A,bool,P,X)) ) ) ).

% ospec
tff(fact_7628_option_Oset__sel,axiom,
    ! [A: $tType,A3: option(A)] :
      ( ( A3 != none(A) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(option(A),A,the2(A),A3)),aa(option(A),set(A),set_option(A),A3))) ) ).

% option.set_sel
tff(fact_7629_option_Osimps_I15_J,axiom,
    ! [A: $tType,X23: A] : aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X23)) = aa(set(A),set(A),insert(A,X23),bot_bot(set(A))) ).

% option.simps(15)
tff(fact_7630_option_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),A3),B2))
    <=> ? [Z5: option(product_prod(A,B))] :
          ( pp(aa(set(option(product_prod(A,B))),bool,aa(option(product_prod(A,B)),fun(set(option(product_prod(A,B))),bool),member(option(product_prod(A,B))),Z5),collect(option(product_prod(A,B)),aTP_Lamp_agx(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),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)),Z5) = 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)),Z5) = B2 ) ) ) ).

% option.in_rel
tff(fact_7631_length__splice,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(list(A),nat,size_size(list(A)),splice(A,Xs,Ys2)) = 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)),Ys2)) ).

% length_splice
tff(fact_7632_rel__option__None1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P),none(A)),X))
    <=> ( X = none(B) ) ) ).

% rel_option_None1
tff(fact_7633_rel__option__None2,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),X: option(A)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P),X),none(B)))
    <=> ( X = none(A) ) ) ).

% rel_option_None2
tff(fact_7634_rel__option__reflI,axiom,
    ! [A: $tType,Y2: option(A),P: fun(A,fun(A,bool))] :
      ( ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(option(A),set(A),set_option(A),Y2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P,X2),X2)) )
     => pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),P),Y2),Y2)) ) ).

% rel_option_reflI
tff(fact_7635_option_Orel__refl__strong,axiom,
    ! [A: $tType,X: option(A),Ra: fun(A,fun(A,bool))] :
      ( ! [Z3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(option(A),set(A),set_option(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),Ra,Z3),Z3)) )
     => pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),Ra),X),X)) ) ).

% option.rel_refl_strong
tff(fact_7636_option_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: option(A),Y2: option(B),Ra: fun(A,fun(B,bool))] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y2))
     => ( ! [Z3: A,Yb: B] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(option(A),set(A),set_option(A),X)))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Yb),aa(option(B),set(B),set_option(B),Y2)))
             => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,Z3),Yb))
               => pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z3),Yb)) ) ) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra),X),Y2)) ) ) ).

% option.rel_mono_strong
tff(fact_7637_option_Orel__cong,axiom,
    ! [A: $tType,B: $tType,X: option(A),Ya: option(A),Y2: option(B),Xa: option(B),R2: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
      ( ( X = Ya )
     => ( ( Y2 = Xa )
       => ( ! [Z3: A,Yb: B] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(option(A),set(A),set_option(A),Ya)))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Yb),aa(option(B),set(B),set_option(B),Xa)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,Z3),Yb))
                <=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z3),Yb)) ) ) )
         => ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y2))
          <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra),Ya),Xa)) ) ) ) ) ).

% option.rel_cong
tff(fact_7638_rel__option__inf,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,fun(B,bool)),B5: fun(A,fun(B,bool))] : aa(fun(option(A),fun(option(B),bool)),fun(option(A),fun(option(B),bool)),aa(fun(option(A),fun(option(B),bool)),fun(fun(option(A),fun(option(B),bool)),fun(option(A),fun(option(B),bool))),inf_inf(fun(option(A),fun(option(B),bool))),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A4)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),B5)) = aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A4),B5)) ).

% rel_option_inf
tff(fact_7639_rel__option__iff,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: option(A),Y2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y2))
    <=> pp(aa(product_prod(option(A),option(B)),bool,product_case_prod(option(A),option(B),bool,aTP_Lamp_aha(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),R2)),aa(option(B),product_prod(option(A),option(B)),product_Pair(option(A),option(B),X),Y2))) ) ).

% rel_option_iff
tff(fact_7640_option_Ocase__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,S3: fun(C,fun(D,bool)),R2: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,D),fun(option(B),D))),bool,aa(fun(C,fun(fun(A,C),fun(option(A),C))),fun(fun(D,fun(fun(B,D),fun(option(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,C),fun(option(A),C)),fun(fun(B,D),fun(option(B),D)),S3,bNF_rel_fun(fun(A,C),fun(B,D),fun(option(A),C),fun(option(B),D),bNF_rel_fun(A,B,C,D,R2,S3),bNF_rel_fun(option(A),option(B),C,D,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),S3))),case_option(C,A)),case_option(D,B))) ).

% option.case_transfer
tff(fact_7641_option_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),Ra))
     => pp(aa(fun(option(A),fun(option(B),bool)),bool,aa(fun(option(A),fun(option(B),bool)),fun(fun(option(A),fun(option(B),bool)),bool),ord_less_eq(fun(option(A),fun(option(B),bool))),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra))) ) ).

% option.rel_mono
tff(fact_7642_option_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(option(B),bool),bool,aa(fun(option(A),bool),fun(fun(option(B),bool),bool),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),fequal(bool)),aTP_Lamp_ahb(option(A),bool)),aTP_Lamp_ahc(option(B),bool))) ).

% option.disc_transfer(2)
tff(fact_7643_option_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(option(B),bool),bool,aa(fun(option(A),bool),fun(fun(option(B),bool),bool),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),fequal(bool)),aTP_Lamp_ahd(option(A),bool)),aTP_Lamp_ahe(option(B),bool))) ).

% option.disc_transfer(1)
tff(fact_7644_option_Octr__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(B,option(B)),bool,aa(fun(A,option(A)),fun(fun(B,option(B)),bool),bNF_rel_fun(A,B,option(A),option(B),R2,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2)),some(A)),some(B))) ).

% option.ctr_transfer(2)
tff(fact_7645_option_Orel__inject_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X23: A,Y23: B] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),aa(A,option(A),some(A),X23)),aa(B,option(B),some(B),Y23)))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),R2,X23),Y23)) ) ).

% option.rel_inject(2)
tff(fact_7646_option_Orel__intros_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X23: A,Y23: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X23),Y23))
     => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),aa(A,option(A),some(A),X23)),aa(B,option(B),some(B),Y23))) ) ).

% option.rel_intros(2)
tff(fact_7647_option__rel__Some1,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),X: A,Y2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A4),aa(A,option(A),some(A),X)),Y2))
    <=> ? [Y7: B] :
          ( ( Y2 = aa(B,option(B),some(B),Y7) )
          & pp(aa(B,bool,aa(A,fun(B,bool),A4,X),Y7)) ) ) ).

% option_rel_Some1
tff(fact_7648_option__rel__Some2,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),X: option(A),Y2: B] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A4),X),aa(B,option(B),some(B),Y2)))
    <=> ? [X9: A] :
          ( ( X = aa(A,option(A),some(A),X9) )
          & pp(aa(B,bool,aa(A,fun(B,bool),A4,X9),Y2)) ) ) ).

% option_rel_Some2
tff(fact_7649_option_Octr__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),none(A)),none(B))) ).

% option.ctr_transfer(1)
tff(fact_7650_option_Orel__induct,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: option(A),Y2: option(B),Q: fun(option(A),fun(option(B),bool))] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y2))
     => ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,none(A)),none(B)))
       => ( ! [A24: A,B22: B] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),R2,A24),B22))
             => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,aa(A,option(A),some(A),A24)),aa(B,option(B),some(B),B22))) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,X),Y2)) ) ) ) ).

% option.rel_induct
tff(fact_7651_option_Orel__cases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),A3),B2))
     => ( ( ( A3 = none(A) )
         => ( B2 != none(B) ) )
       => ~ ! [X2: A] :
              ( ( A3 = aa(A,option(A),some(A),X2) )
             => ! [Y3: B] :
                  ( ( B2 = aa(B,option(B),some(B),Y3) )
                 => ~ pp(aa(B,bool,aa(A,fun(B,bool),R2,X2),Y3)) ) ) ) ) ).

% option.rel_cases
tff(fact_7652_option_Orel__distinct_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),Y23: B] : ~ pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),none(A)),aa(B,option(B),some(B),Y23))) ).

% option.rel_distinct(1)
tff(fact_7653_option_Orel__distinct_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),Y23: A] : ~ pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),aa(A,option(A),some(A),Y23)),none(B))) ).

% option.rel_distinct(2)
tff(fact_7654_option_Orel__sel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),A3),B2))
    <=> ( ( ( A3 = none(A) )
        <=> ( B2 = none(B) ) )
        & ( ( A3 != none(A) )
         => ( ( B2 != none(B) )
           => pp(aa(B,bool,aa(A,fun(B,bool),R2,aa(option(A),A,the2(A),A3)),aa(option(B),B,the2(B),B2))) ) ) ) ) ).

% option.rel_sel
tff(fact_7655_option_Orel__eq,axiom,
    ! [A: $tType] : aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),fequal(A)) = fequal(option(A)) ).

% option.rel_eq
tff(fact_7656_option_Orel__refl,axiom,
    ! [B: $tType,Ra: fun(B,fun(B,bool)),X: option(B)] :
      ( ! [X2: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra,X2),X2))
     => pp(aa(option(B),bool,aa(option(B),fun(option(B),bool),aa(fun(B,fun(B,bool)),fun(option(B),fun(option(B),bool)),rel_option(B,B),Ra),X),X)) ) ).

% option.rel_refl
tff(fact_7657_option_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: option(A),G: fun(B,C),Y2: option(B)] :
      ( pp(aa(option(C),bool,aa(option(A),fun(option(C),bool),aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),Sa),X),aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),G),Y2)))
    <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ahf(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y2)) ) ).

% option.rel_map(2)
tff(fact_7658_option_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X: option(A),Y2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(C),fun(option(B),bool),aa(fun(C,fun(B,bool)),fun(option(C),fun(option(B),bool)),rel_option(C,B),Sb),aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),I),X)),Y2))
    <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ahg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X),Y2)) ) ).

% option.rel_map(1)
tff(fact_7659_option_Orec__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,S3: fun(C,fun(D,bool)),R2: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,D),fun(option(B),D))),bool,aa(fun(C,fun(fun(A,C),fun(option(A),C))),fun(fun(D,fun(fun(B,D),fun(option(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,C),fun(option(A),C)),fun(fun(B,D),fun(option(B),D)),S3,bNF_rel_fun(fun(A,C),fun(B,D),fun(option(A),C),fun(option(B),D),bNF_rel_fun(A,B,C,D,R2,S3),bNF_rel_fun(option(A),option(B),C,D,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),S3))),rec_option(C,A)),rec_option(D,B))) ).

% option.rec_transfer
tff(fact_7660_option_Omap__transfer,axiom,
    ! [A: $tType,B: $tType,F: $tType,E5: $tType,Rb: fun(A,fun(E5,bool)),Sd: fun(B,fun(F,bool))] : pp(aa(fun(fun(E5,F),fun(option(E5),option(F))),bool,aa(fun(fun(A,B),fun(option(A),option(B))),fun(fun(fun(E5,F),fun(option(E5),option(F))),bool),bNF_rel_fun(fun(A,B),fun(E5,F),fun(option(A),option(B)),fun(option(E5),option(F)),bNF_rel_fun(A,E5,B,F,Rb,Sd),bNF_rel_fun(option(A),option(E5),option(B),option(F),aa(fun(A,fun(E5,bool)),fun(option(A),fun(option(E5),bool)),rel_option(A,E5),Rb),aa(fun(B,fun(F,bool)),fun(option(B),fun(option(F),bool)),rel_option(B,F),Sd))),map_option(A,B)),map_option(E5,F))) ).

% option.map_transfer
tff(fact_7661_option_Orel__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,bool)),Sc: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool))),bool,aa(fun(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool))),fun(fun(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool))),bool),bNF_rel_fun(fun(A,fun(B,bool)),fun(C,fun(D,bool)),fun(option(A),fun(option(B),bool)),fun(option(C),fun(option(D),bool)),bNF_rel_fun(A,C,fun(B,bool),fun(D,bool),Sa,bNF_rel_fun(B,D,bool,bool,Sc,fequal(bool))),bNF_rel_fun(option(A),option(C),fun(option(B),bool),fun(option(D),bool),aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),Sa),bNF_rel_fun(option(B),option(D),bool,bool,aa(fun(B,fun(D,bool)),fun(option(B),fun(option(D),bool)),rel_option(B,D),Sc),fequal(bool)))),rel_option(A,B)),rel_option(C,D))) ).

% option.rel_transfer
tff(fact_7662_option_Orel__transp,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( transp(A,R2)
     => transp(option(A),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),R2)) ) ).

% option.rel_transp
tff(fact_7663_option_OQuotient,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T5: fun(A,fun(B,bool))] :
      ( quotient(A,B,R2,Abs,Rep,T5)
     => quotient(option(A),option(B),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),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,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),T5)) ) ).

% option.Quotient
tff(fact_7664_option__bind__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] : pp(aa(fun(option(B),fun(fun(B,option(D)),option(D))),bool,aa(fun(option(A),fun(fun(A,option(C)),option(C))),fun(fun(option(B),fun(fun(B,option(D)),option(D))),bool),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,bool)),fun(option(A),fun(option(B),bool)),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,bool)),fun(option(C),fun(option(D),bool)),rel_option(C,D),B5)),aa(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool)),rel_option(C,D),B5))),bind(A,C)),bind(B,D))) ).

% option_bind_transfer
tff(fact_7665_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_ahh(list(A),product_prod(nat,list(A)))) ).

% lenlex_def
tff(fact_7666_pairwise__alt,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),S3: set(A)] :
      ( pairwise(A,R2,S3)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),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,X4),bot_bot(set(A))))))
             => pp(aa(A,bool,aa(A,fun(A,bool),R2,X4),Xa3)) ) ) ) ).

% pairwise_alt
tff(fact_7667_pred__option__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(option(B),bool)),bool,aa(fun(fun(A,bool),fun(option(A),bool)),fun(fun(fun(B,bool),fun(option(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(option(A),bool),fun(option(B),bool),bNF_rel_fun(A,B,bool,bool,A4,fequal(bool)),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A4),fequal(bool))),pred_option(A)),pred_option(B))) ).

% pred_option_parametric
tff(fact_7668_option_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(option(B),bool)),bool,aa(fun(fun(A,bool),fun(option(A),bool)),fun(fun(fun(B,bool),fun(option(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(option(A),bool),fun(option(B),bool),bNF_rel_fun(A,B,bool,bool,R2,fequal(bool)),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),fequal(bool))),pred_option(A)),pred_option(B))) ).

% option.pred_transfer
tff(fact_7669_option_Opred__inject_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),aa(A,option(A),some(A),A3)))
    <=> pp(aa(A,bool,P,A3)) ) ).

% option.pred_inject(2)
tff(fact_7670_option_Opred__mono,axiom,
    ! [A: $tType,P: fun(A,bool),Pa: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Pa))
     => pp(aa(fun(option(A),bool),bool,aa(fun(option(A),bool),fun(fun(option(A),bool),bool),ord_less_eq(fun(option(A),bool)),aa(fun(A,bool),fun(option(A),bool),pred_option(A),P)),aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa))) ) ).

% option.pred_mono
tff(fact_7671_option_Opred__inject_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),none(A))) ).

% option.pred_inject(1)
tff(fact_7672_option_Opred__True,axiom,
    ! [A: $tType,X3: option(A)] : pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aTP_Lamp_ne(A,bool)),X3)) ).

% option.pred_True
tff(fact_7673_option_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,X: option(A),Ya: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_ahi(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G)),Ya))
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),X) = 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_7674_option_Opred__cong,axiom,
    ! [A: $tType,X: option(A),Ya: option(A),P: fun(A,bool),Pa: fun(A,bool)] :
      ( ( X = Ya )
     => ( ! [Z3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(option(A),set(A),set_option(A),Ya)))
           => ( pp(aa(A,bool,P,Z3))
            <=> pp(aa(A,bool,Pa,Z3)) ) )
       => ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X))
        <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa),Ya)) ) ) ) ).

% option.pred_cong
tff(fact_7675_option_Opred__mono__strong,axiom,
    ! [A: $tType,P: fun(A,bool),X: option(A),Pa: fun(A,bool)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X))
     => ( ! [Z3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(option(A),set(A),set_option(A),X)))
           => ( pp(aa(A,bool,P,Z3))
             => pp(aa(A,bool,Pa,Z3)) ) )
       => pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa),X)) ) ) ).

% option.pred_mono_strong
tff(fact_7676_option_Opred__set,axiom,
    ! [A: $tType,P: fun(A,bool),X3: option(A)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X3))
    <=> ! [Xa3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(option(A),set(A),set_option(A),X3)))
         => pp(aa(A,bool,P,Xa3)) ) ) ).

% option.pred_set
tff(fact_7677_option_Opred__map,axiom,
    ! [B: $tType,A: $tType,Q: fun(B,bool),F2: fun(A,B),X: option(A)] :
      ( pp(aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),pred_option(B),Q),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F2),X)))
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aa(fun(A,B),fun(A,bool),comp(B,bool,A,Q),F2)),X)) ) ).

% option.pred_map
tff(fact_7678_map__le__imp__upd__le,axiom,
    ! [A: $tType,B: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X: A,Y2: B] :
      ( map_le(A,B,M1,M22)
     => map_le(A,B,fun_upd(A,option(B),M1,X,none(B)),fun_upd(A,option(B),M22,X,aa(B,option(B),some(B),Y2))) ) ).

% map_le_imp_upd_le
tff(fact_7679_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_7680_lcm_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% lcm.bottom_right_bottom
tff(fact_7681_lcm_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% lcm.bottom_left_bottom
tff(fact_7682_lcm__neg1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm_neg1
tff(fact_7683_lcm__neg2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm_neg2
tff(fact_7684_lcm__0__iff__int,axiom,
    ! [M2: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M2),N) = zero_zero(int) )
    <=> ( ( M2 = zero_zero(int) )
        | ( N = zero_zero(int) ) ) ) ).

% lcm_0_iff_int
tff(fact_7685_lcm__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,N: 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),N))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(num,A,numeral_numeral(A),N)) ) ).

% lcm_neg_numeral_2
tff(fact_7686_lcm__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: 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),N))),A3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(num,A,numeral_numeral(A),N)),A3) ) ).

% lcm_neg_numeral_1
tff(fact_7687_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) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).

% lcm_eq_1_iff
tff(fact_7688_lcm__1__iff__int,axiom,
    ! [M2: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M2),N) = one_one(int) )
    <=> ( ( ( M2 = one_one(int) )
          | ( M2 = aa(int,int,uminus_uminus(int),one_one(int)) ) )
        & ( ( N = one_one(int) )
          | ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% lcm_1_iff_int
tff(fact_7689_lcm__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7690_lcm__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7691_lcm__neg1__int,axiom,
    ! [X: int,Y2: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),Y2) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y2) ).

% lcm_neg1_int
tff(fact_7692_lcm__neg2__int,axiom,
    ! [X: int,Y2: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),aa(int,int,uminus_uminus(int),Y2)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y2) ).

% lcm_neg2_int
tff(fact_7693_map__le__empty,axiom,
    ! [B: $tType,A: $tType,G: fun(A,option(B))] : map_le(A,B,aTP_Lamp_acf(A,option(B)),G) ).

% map_le_empty
tff(fact_7694_upd__None__map__le,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),X: A] : map_le(A,B,fun_upd(A,option(B),F2,X,none(B)),F2) ).

% upd_None_map_le
tff(fact_7695_zero__eq__lcm__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_eq_lcm_iff
tff(fact_7696_lcm__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% lcm_eq_0_iff
tff(fact_7697_lcm__pos__int,axiom,
    ! [M2: int,N: int] :
      ( ( M2 != zero_zero(int) )
     => ( ( N != zero_zero(int) )
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M2),N))) ) ) ).

% lcm_pos_int
tff(fact_7698_lcm__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7699_lcm__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7700_lcm__unique__int,axiom,
    ! [D2: int,A3: int,B2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D2))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),D2))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),D2))
        & ! [E3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),E3))
              & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),E3)) )
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),E3)) ) )
    <=> ( D2 = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A3),B2) ) ) ).

% lcm_unique_int
tff(fact_7701_lcm__ge__0__int,axiom,
    ! [X: int,Y2: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y2))) ).

% lcm_ge_0_int
tff(fact_7702_lcm__cases__int,axiom,
    ! [X: int,Y2: int,P: fun(int,bool)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y2))) ) )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y2),zero_zero(int)))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),aa(int,int,uminus_uminus(int),Y2)))) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y2))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),Y2))) ) )
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y2),zero_zero(int)))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y2)))) ) )
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y2))) ) ) ) ) ).

% lcm_cases_int
tff(fact_7703_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list(int)] : gcd_Lcm(int,aa(list(int),set(int),set2(int),Xs)) = fold(int,int,gcd_lcm(int),Xs,one_one(int)) ).

% Lcm_int_set_eq_fold
tff(fact_7704_Lcm__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite(A),A4))
           => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = finite_fold(A,A,gcd_lcm(A),one_one(A),A4) ) )
          & ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
           => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = zero_zero(A) ) ) ) ) ).

% Lcm_fin.eq_fold
tff(fact_7705_lcm__0__iff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M2),N) = zero_zero(nat) )
    <=> ( ( M2 = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% lcm_0_iff_nat
tff(fact_7706_lcm__1__iff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M2),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% lcm_1_iff_nat
tff(fact_7707_lcm__int__int__eq,axiom,
    ! [M2: nat,N: nat] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M2),N)) ).

% lcm_int_int_eq
tff(fact_7708_Lcm__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,top_top(set(A))) = zero_zero(A) ) ) ).

% Lcm_UNIV
tff(fact_7709_Lcm__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_empty
tff(fact_7710_Lcm__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( gcd_Lcm(A,A4) = one_one(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X4),one_one(A))) ) ) ) ).

% Lcm_1_iff
tff(fact_7711_Lcm__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = zero_zero(A) ) ) ) ).

% Lcm_fin.infinite
tff(fact_7712_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_7713_is__unit__Lcm__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7714_lcm__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ).

% lcm_nat_abs_right_eq
tff(fact_7715_lcm__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ).

% lcm_nat_abs_left_eq
tff(fact_7716_lcm__nat__def,axiom,
    ! [X: nat,Y2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),X),Y2) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y2)) ).

% lcm_nat_def
tff(fact_7717_lcm__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M2),N))) ) ) ).

% lcm_pos_nat
tff(fact_7718_Lcm__int__greater__eq__0,axiom,
    ! [K6: set(int)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),gcd_Lcm(int,K6))) ).

% Lcm_int_greater_eq_0
tff(fact_7719_Lcm__eq__0__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A4))
         => ( gcd_Lcm(A,A4) = zero_zero(A) ) ) ) ).

% Lcm_eq_0_I
tff(fact_7720_Lcm__no__multiple,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ! [M3: A] :
              ( ( M3 != zero_zero(A) )
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A4))
                  & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),M3)) ) )
         => ( gcd_Lcm(A,A4) = zero_zero(A) ) ) ) ).

% Lcm_no_multiple
tff(fact_7721_Lcm__0__iff_H,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( gcd_Lcm(A,A4) = zero_zero(A) )
        <=> ~ ? [L3: A] :
                ( ( L3 != zero_zero(A) )
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X4),L3)) ) ) ) ) ).

% Lcm_0_iff'
tff(fact_7722_Lcm__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = zero_zero(A) )
          <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A4)) ) ) ) ).

% Lcm_fin_0_iff
tff(fact_7723_Lcm__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A4))
         => ( ( gcd_Lcm(A,A4) = zero_zero(A) )
          <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A4)) ) ) ) ).

% Lcm_0_iff
tff(fact_7724_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) )
        <=> ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A4))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X4),one_one(A))) )
            & pp(aa(set(A),bool,finite_finite(A),A4)) ) ) ) ).

% Lcm_fin_1_iff
tff(fact_7725_lcm__code__integer,axiom,
    ! [A3: code_integer,B2: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_lcm(code_integer),A3),B2) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),A3)),aa(code_integer,code_integer,abs_abs(code_integer),B2))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),A3),B2)) ).

% lcm_code_integer
tff(fact_7726_Lcm__no__units,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] : gcd_Lcm(A,A4) = gcd_Lcm(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),collect(A,aTP_Lamp_ahj(A,bool)))) ) ).

% Lcm_no_units
tff(fact_7727_lcm__int__def,axiom,
    ! [X: int,Y2: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y2) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),X))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y2)))) ).

% lcm_int_def
tff(fact_7728_Lcm__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),insert(A,A3),A4)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(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),bot_bot(set(A)))))) ) ).

% Lcm_fin.insert_remove
tff(fact_7729_Lcm__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A4))
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(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),bot_bot(set(A)))))) ) ) ) ).

% Lcm_fin.remove
tff(fact_7730_Lcm__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : gcd_Lcm(A,aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_lcm(A),Xs,one_one(A)) ) ).

% Lcm_set_eq_fold
tff(fact_7731_Lcm__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_lcm(A),Xs,one_one(A)) ) ).

% Lcm_fin.set_eq_fold
tff(fact_7732_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_7733_Lcm__int__def,axiom,
    ! [K6: set(int)] : gcd_Lcm(int,K6) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)),K6))) ).

% Lcm_int_def
tff(fact_7734_Lcm__int__eq,axiom,
    ! [N6: set(nat)] : gcd_Lcm(int,image(nat,int,semiring_1_of_nat(int),N6)) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,N6)) ).

% Lcm_int_eq
tff(fact_7735_Lcm__eq__0__I__nat,axiom,
    ! [A4: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4))
     => ( gcd_Lcm(nat,A4) = zero_zero(nat) ) ) ).

% Lcm_eq_0_I_nat
tff(fact_7736_Lcm__0__iff__nat,axiom,
    ! [A4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A4))
     => ( ( gcd_Lcm(nat,A4) = zero_zero(nat) )
      <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A4)) ) ) ).

% Lcm_0_iff_nat
tff(fact_7737_Lcm__nat__empty,axiom,
    gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).

% Lcm_nat_empty
tff(fact_7738_Lcm__nat__infinite,axiom,
    ! [M10: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite(nat),M10))
     => ( gcd_Lcm(nat,M10) = zero_zero(nat) ) ) ).

% Lcm_nat_infinite
tff(fact_7739_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Lcm(nat,aa(list(nat),set(nat),set2(nat),Xs)) = fold(nat,nat,gcd_lcm(nat),Xs,one_one(nat)) ).

% Lcm_nat_set_eq_fold
tff(fact_7740_Lcm__eq__Max__nat,axiom,
    ! [M10: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),M10))
     => ( ( M10 != bot_bot(set(nat)) )
       => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M10))
         => ( ! [M3: nat,N2: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),M3),M10))
               => ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),M10))
                 => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M3),N2)),M10)) ) )
           => ( gcd_Lcm(nat,M10) = lattic643756798349783984er_Max(nat,M10) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_7741_Lcm__nat__def,axiom,
    ! [M10: set(nat)] :
      ( ( pp(aa(set(nat),bool,finite_finite(nat),M10))
       => ( gcd_Lcm(nat,M10) = lattic5214292709420241887eutr_F(nat,gcd_lcm(nat),one_one(nat),M10) ) )
      & ( ~ pp(aa(set(nat),bool,finite_finite(nat),M10))
       => ( gcd_Lcm(nat,M10) = zero_zero(nat) ) ) ) ).

% Lcm_nat_def
tff(fact_7742_Lcm__coprime_H,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),A4) != zero_zero(nat) )
         => ( ! [A6: A,B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),A4))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),A4))
                 => ( ( A6 != B4 )
                   => algebr8660921524188924756oprime(A,A6,B4) ) ) )
           => ( gcd_Lcm(A,A4) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_ahk(A,A)),A4)) ) ) ) ) ).

% Lcm_coprime'
tff(fact_7743_normalize__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,normal6383669964737779283malize(A),A3)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% normalize_idem
tff(fact_7744_normalize__0,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).

% normalize_0
tff(fact_7745_normalize__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% normalize_eq_0_iff
tff(fact_7746_lcm_Onormalize__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).

% lcm.normalize_bottom
tff(fact_7747_normalize__mult__normalize__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% normalize_mult_normalize_left
tff(fact_7748_normalize__mult__normalize__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,normal6383669964737779283malize(A),B2))) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% normalize_mult_normalize_right
tff(fact_7749_gcd_Onormalize__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).

% gcd.normalize_bottom
tff(fact_7750_normalize__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).

% normalize_1
tff(fact_7751_dvd__normalize__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,normal6383669964737779283malize(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_normalize_iff
tff(fact_7752_normalize__dvd__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% normalize_dvd_iff
tff(fact_7753_coprime__normalize__right__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,aa(A,A,normal6383669964737779283malize(A),B2))
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_normalize_right_iff
tff(fact_7754_coprime__normalize__left__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,normal6383669964737779283malize(A),A3),B2)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_normalize_left_iff
tff(fact_7755_gcd_Otop__right__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),zero_zero(A)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% gcd.top_right_normalize
tff(fact_7756_gcd_Otop__left__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),zero_zero(A)),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% gcd.top_left_normalize
tff(fact_7757_lcm_Otop__right__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),one_one(A)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% lcm.top_right_normalize
tff(fact_7758_lcm_Otop__left__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),one_one(A)),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% lcm.top_left_normalize
tff(fact_7759_normalize__mult__unit__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).

% normalize_mult_unit_left
tff(fact_7760_normalize__mult__unit__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ) ) ).

% normalize_mult_unit_right
tff(fact_7761_normalize__idem__imp__is__unit__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
          <=> ( A3 = one_one(A) ) ) ) ) ).

% normalize_idem_imp_is_unit_iff
tff(fact_7762_is__unit__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) ) ) ) ).

% is_unit_normalize
tff(fact_7763_normalize__1__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% normalize_1_iff
tff(fact_7764_associated__unit,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).

% associated_unit
tff(fact_7765_associated__iff__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ) ).

% associated_iff_dvd
tff(fact_7766_associated__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
             => ( ( aa(A,A,normal6383669964737779283malize(A),B2) = B2 )
               => ( A3 = B2 ) ) ) ) ) ) ).

% associated_eqI
tff(fact_7767_associatedD2,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% associatedD2
tff(fact_7768_associatedD1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% associatedD1
tff(fact_7769_associatedI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ) ).

% associatedI
tff(fact_7770_dvd__normalize__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( aa(A,A,normal6383669964737779283malize(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,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ) ) ).

% dvd_normalize_div
tff(fact_7771_normalize__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(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,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).

% normalize_mult
tff(fact_7772_lcm__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,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_gcd(A),A3),B2))) ) ).

% lcm_gcd
tff(fact_7773_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde6485984586167503788ce_set(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).

% Gcd_fin.bounded_quasi_semilattice_set_axioms
tff(fact_7774_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde6485984586167503788ce_set(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).

% Lcm_fin.bounded_quasi_semilattice_set_axioms
tff(fact_7775_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) = aa(A,A,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_7776_semilattice__neutr__set_Oremove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A4: set(A),X: A] :
      ( lattic5652469242046573047tr_set(A,F2,Z)
     => ( pp(aa(set(A),bool,finite_finite(A),A4))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
         => ( lattic5214292709420241887eutr_F(A,F2,Z,A4) = aa(A,A,aa(A,fun(A,A),F2,X),lattic5214292709420241887eutr_F(A,F2,Z,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,X),bot_bot(set(A)))))) ) ) ) ) ).

% semilattice_neutr_set.remove
tff(fact_7777_semilattice__neutr__set_Oinsert__remove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A4: set(A),X: A] :
      ( lattic5652469242046573047tr_set(A,F2,Z)
     => ( pp(aa(set(A),bool,finite_finite(A),A4))
       => ( lattic5214292709420241887eutr_F(A,F2,Z,aa(set(A),set(A),insert(A,X),A4)) = aa(A,A,aa(A,fun(A,A),F2,X),lattic5214292709420241887eutr_F(A,F2,Z,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,X),bot_bot(set(A)))))) ) ) ) ).

% semilattice_neutr_set.insert_remove
tff(fact_7778_lcm_Obounded__quasi__semilattice__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde8507323023520639062attice(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).

% lcm.bounded_quasi_semilattice_axioms
tff(fact_7779_gcd_Obounded__quasi__semilattice__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde8507323023520639062attice(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).

% gcd.bounded_quasi_semilattice_axioms
tff(fact_7780_normalize__div,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,normal6383669964737779283malize(A),A3)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,A3)) ) ).

% normalize_div
tff(fact_7781_unit__factor__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( unit_f5069060285200089521factor(A,aa(A,A,normal6383669964737779283malize(A),A3)) = one_one(A) ) ) ) ).

% unit_factor_normalize
tff(fact_7782_unit__factor__simps_I1_J,axiom,
    unit_f5069060285200089521factor(nat,zero_zero(nat)) = zero_zero(nat) ).

% unit_factor_simps(1)
tff(fact_7783_unit__factor__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : unit_f5069060285200089521factor(A,unit_f5069060285200089521factor(A,A3)) = unit_f5069060285200089521factor(A,A3) ) ).

% unit_factor_idem
tff(fact_7784_unit__factor__simps_I2_J,axiom,
    ! [N: nat] : unit_f5069060285200089521factor(nat,aa(nat,nat,suc,N)) = one_one(nat) ).

% unit_factor_simps(2)
tff(fact_7785_unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( unit_f5069060285200089521factor(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% unit_factor_eq_0_iff
tff(fact_7786_unit__factor__0,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ( unit_f5069060285200089521factor(A,zero_zero(A)) = zero_zero(A) ) ) ).

% unit_factor_0
tff(fact_7787_unit__factor__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).

% unit_factor_1
tff(fact_7788_unit__factor__mult__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A3)),aa(A,A,normal6383669964737779283malize(A),A3)) = A3 ) ).

% unit_factor_mult_normalize
tff(fact_7789_normalize__mult__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),unit_f5069060285200089521factor(A,A3)) = A3 ) ).

% normalize_mult_unit_factor
tff(fact_7790_div__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,normal6383669964737779283malize(A),A3)) = unit_f5069060285200089521factor(A,A3) ) ).

% div_normalize
tff(fact_7791_div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),unit_f5069060285200089521factor(A,A3)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% div_unit_factor
tff(fact_7792_inv__unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,A3)) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inv_unit_factor_eq_0_iff
tff(fact_7793_mult__one__div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),unit_f5069060285200089521factor(A,B2)) ) ).

% mult_one_div_unit_factor
tff(fact_7794_unit__factor__mult__unit__left,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( unit_f5069060285200089521factor(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),unit_f5069060285200089521factor(A,B2)) ) ) ) ).

% unit_factor_mult_unit_left
tff(fact_7795_unit__factor__mult__unit__right,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( unit_f5069060285200089521factor(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),unit_f5069060285200089521factor(A,B2)),A3) ) ) ) ).

% unit_factor_mult_unit_right
tff(fact_7796_unit__factor__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( ( ( A3 = zero_zero(A) )
              | ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = zero_zero(A) ) )
          & ( ~ ( ( A3 = zero_zero(A) )
                | ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = one_one(A) ) ) ) ) ).

% unit_factor_lcm
tff(fact_7797_normalize__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,normal6383669964737779283malize(A),unit_f5069060285200089521factor(A,A3)) = one_one(A) ) ) ) ).

% normalize_unit_factor
tff(fact_7798_normalize__unit__factor__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => ( ( unit_f5069060285200089521factor(A,A3) = unit_f5069060285200089521factor(A,B2) )
           => ( A3 = B2 ) ) ) ) ).

% normalize_unit_factor_eqI
tff(fact_7799_unit__factor__is__unit,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),one_one(A))) ) ) ).

% unit_factor_is_unit
tff(fact_7800_unit__factor__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A3: A,B2: A] : unit_f5069060285200089521factor(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),unit_f5069060285200089521factor(A,A3)),unit_f5069060285200089521factor(A,B2)) ) ).

% unit_factor_mult
tff(fact_7801_unit__factor__self,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),A3)) ) ).

% unit_factor_self
tff(fact_7802_dvd__unit__factor__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( unit_f5069060285200089521factor(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),unit_f5069060285200089521factor(A,A3)),unit_f5069060285200089521factor(A,B2)) ) ) ) ).

% dvd_unit_factor_div
tff(fact_7803_unit__factor__nat__def,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( unit_f5069060285200089521factor(nat,N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( unit_f5069060285200089521factor(nat,N) = one_one(nat) ) ) ) ).

% unit_factor_nat_def
tff(fact_7804_unit__factor__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),B2)) ) ) ).

% unit_factor_dvd
tff(fact_7805_is__unit__unit__factor,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( unit_f5069060285200089521factor(A,A3) = A3 ) ) ) ).

% is_unit_unit_factor
tff(fact_7806_unit__factor__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( ( ( A3 = zero_zero(A) )
              & ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = zero_zero(A) ) )
          & ( ~ ( ( A3 = zero_zero(A) )
                & ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = one_one(A) ) ) ) ) ).

% unit_factor_gcd
tff(fact_7807_coprime__crossproduct_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [B2: A,D2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( unit_f5069060285200089521factor(A,B2) = unit_f5069060285200089521factor(A,D2) )
           => ( algebr8660921524188924756oprime(A,A3,B2)
             => ( algebr8660921524188924756oprime(A,C2,D2)
               => ( ( 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) )
                <=> ( ( A3 = C2 )
                    & ( B2 = D2 ) ) ) ) ) ) ) ) ).

% coprime_crossproduct'
tff(fact_7808_unit__factor__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( ( gcd_Lcm(A,A4) = zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A4)) = zero_zero(A) ) )
          & ( ( gcd_Lcm(A,A4) != zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A4)) = one_one(A) ) ) ) ) ).

% unit_factor_Lcm
tff(fact_7809_unit__factor__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( ( gcd_Gcd(A,A4) = zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A4)) = zero_zero(A) ) )
          & ( ( gcd_Gcd(A,A4) != zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A4)) = one_one(A) ) ) ) ) ).

% unit_factor_Gcd
tff(fact_7810_normalize__idem__imp__unit__factor__eq,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
         => ( unit_f5069060285200089521factor(A,A3) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ) ) ).

% normalize_idem_imp_unit_factor_eq
tff(fact_7811_unit__factor__Lcm__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] : unit_f5069060285200089521factor(A,aa(set(A),A,semiring_gcd_Lcm_fin(A),A4)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A4)),zero_zero(A)))) ) ).

% unit_factor_Lcm_fin
tff(fact_7812_unit__factor__Gcd__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] : unit_f5069060285200089521factor(A,aa(set(A),A,semiring_gcd_Gcd_fin(A),A4)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A4)),zero_zero(A)))) ) ).

% unit_factor_Gcd_fin
tff(fact_7813_and_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => comm_monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.comm_monoid_axioms
tff(fact_7814_card__Plus__conv__if,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
      ( ( ( pp(aa(set(A),bool,finite_finite(A),A4))
          & pp(aa(set(B),bool,finite_finite(B),B5)) )
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A4,B5)) = 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),B5)) ) )
      & ( ~ ( pp(aa(set(A),bool,finite_finite(A),A4))
            & pp(aa(set(B),bool,finite_finite(B),B5)) )
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A4,B5)) = zero_zero(nat) ) ) ) ).

% card_Plus_conv_if
tff(fact_7815_gcd__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.comm_monoid_axioms
tff(fact_7816_inf__top_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => comm_monoid(A,inf_inf(A),top_top(A)) ) ).

% inf_top.comm_monoid_axioms
tff(fact_7817_semilattice__neutr_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( semilattice_neutr(A,F2,Z)
     => comm_monoid(A,F2,Z) ) ).

% semilattice_neutr.axioms(2)
tff(fact_7818_or_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => comm_monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.comm_monoid_axioms
tff(fact_7819_xor_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => comm_monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.comm_monoid_axioms
tff(fact_7820_max__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.comm_monoid_axioms
tff(fact_7821_comm__monoid_Ocomm__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A3: A] :
      ( comm_monoid(A,F2,Z)
     => ( aa(A,A,aa(A,fun(A,A),F2,A3),Z) = A3 ) ) ).

% comm_monoid.comm_neutral
tff(fact_7822_sup__bot_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => comm_monoid(A,sup_sup(A),bot_bot(A)) ) ).

% sup_bot.comm_monoid_axioms
tff(fact_7823_add_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => comm_monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.comm_monoid_axioms
tff(fact_7824_mult_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => comm_monoid(A,times_times(A),one_one(A)) ) ).

% mult.comm_monoid_axioms
tff(fact_7825_lexn_Osimps_I2_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] : lexn(A,R,aa(nat,nat,suc,N)) = 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),product_case_prod(A,list(A),list(A),cons(A)),product_case_prod(A,list(A),list(A),cons(A))),lex_prod(A,list(A),R,lexn(A,R,N)))),collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_ahl(nat,fun(list(A),fun(list(A),bool)),N)))) ).

% lexn.simps(2)
tff(fact_7826_is__num__normalize_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A,Y2: A] :
          ( neg_numeral_is_num(A,X)
         => ( neg_numeral_is_num(A,Y2)
           => neg_numeral_is_num(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) ) ) ) ).

% is_num_normalize(6)
tff(fact_7827_is__num__add__commute,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A,Y2: A] :
          ( neg_numeral_is_num(A,X)
         => ( neg_numeral_is_num(A,Y2)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y2),X) ) ) ) ) ).

% is_num_add_commute
tff(fact_7828_is__num__add__left__commute,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A,Y2: A,Z: A] :
          ( neg_numeral_is_num(A,X)
         => ( neg_numeral_is_num(A,Y2)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y2),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ) ) ) ).

% is_num_add_left_commute
tff(fact_7829_is__num__normalize_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] :
          ( neg_numeral_is_num(A,X)
         => neg_numeral_is_num(A,aa(A,A,uminus_uminus(A),X)) ) ) ).

% is_num_normalize(5)
tff(fact_7830_is__num__numeral,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_is_num(A,aa(num,A,numeral_numeral(A),K)) ) ).

% is_num_numeral
tff(fact_7831_is__num__normalize_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => neg_numeral_is_num(A,one_one(A)) ) ).

% is_num_normalize(4)
tff(fact_7832_is__num_Ocases,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A] :
          ( neg_numeral_is_num(A,A3)
         => ( ( A3 != one_one(A) )
           => ( ! [X2: A] :
                  ( ( A3 = aa(A,A,uminus_uminus(A),X2) )
                 => ~ neg_numeral_is_num(A,X2) )
             => ~ ! [X2: A,Y3: A] :
                    ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y3) )
                   => ( neg_numeral_is_num(A,X2)
                     => ~ neg_numeral_is_num(A,Y3) ) ) ) ) ) ) ).

% is_num.cases
tff(fact_7833_is__num_Osimps,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A] :
          ( neg_numeral_is_num(A,A3)
        <=> ( ( A3 = one_one(A) )
            | ? [X4: A] :
                ( ( A3 = aa(A,A,uminus_uminus(A),X4) )
                & neg_numeral_is_num(A,X4) )
            | ? [X4: A,Y4: A] :
                ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y4) )
                & neg_numeral_is_num(A,X4)
                & neg_numeral_is_num(A,Y4) ) ) ) ) ).

% is_num.simps
tff(fact_7834_times__num__def,axiom,
    ! [M2: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M2),N) = num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,M2)),aa(num,nat,nat_of_num,N))) ).

% times_num_def
tff(fact_7835_arg__max__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F2: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,Y3)),B2)) )
       => ( pp(aa(A,bool,P,lattices_ord_arg_max(A,nat,F2,P)))
          & ! [Y: A] :
              ( pp(aa(A,bool,P,Y))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,Y)),aa(A,nat,F2,lattices_ord_arg_max(A,nat,F2,P)))) ) ) ) ) ).

% arg_max_nat_lemma
tff(fact_7836_less__eq__num__def,axiom,
    ! [M2: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,nat_of_num,M2)),aa(num,nat,nat_of_num,N))) ) ).

% less_eq_num_def
tff(fact_7837_nat__of__num__code_I1_J,axiom,
    aa(num,nat,nat_of_num,one2) = one_one(nat) ).

% nat_of_num_code(1)
tff(fact_7838_nat__of__num__neq__0,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,X) != zero_zero(nat) ).

% nat_of_num_neq_0
tff(fact_7839_nat__of__num__pos,axiom,
    ! [X: num] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,nat_of_num,X))) ).

% nat_of_num_pos
tff(fact_7840_arg__maxI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [P: fun(A,bool),X: A,F2: fun(A,B),Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y3))) )
           => ( ! [X2: A] :
                  ( pp(aa(A,bool,P,X2))
                 => ( ! [Y: A] :
                        ( pp(aa(A,bool,P,Y))
                       => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y))) )
                   => pp(aa(A,bool,Q,X2)) ) )
             => pp(aa(A,bool,Q,lattices_ord_arg_max(A,B,F2,P))) ) ) ) ) ).

% arg_maxI
tff(fact_7841_nat__of__num__inc,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,inc(X)) = aa(nat,nat,suc,aa(num,nat,nat_of_num,X)) ).

% nat_of_num_inc
tff(fact_7842_arg__max__natI,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F2: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,Y3)),B2)) )
       => pp(aa(A,bool,P,lattices_ord_arg_max(A,nat,F2,P))) ) ) ).

% arg_max_natI
tff(fact_7843_num__eq__iff,axiom,
    ! [X: num,Y2: num] :
      ( ( X = Y2 )
    <=> ( aa(num,nat,nat_of_num,X) = aa(num,nat,nat_of_num,Y2) ) ) ).

% num_eq_iff
tff(fact_7844_nat__of__num__numeral,axiom,
    nat_of_num = numeral_numeral(nat) ).

% nat_of_num_numeral
tff(fact_7845_nat__of__num__inverse,axiom,
    ! [X: num] : num_of_nat(aa(num,nat,nat_of_num,X)) = X ).

% nat_of_num_inverse
tff(fact_7846_less__num__def,axiom,
    ! [M2: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,nat_of_num,M2)),aa(num,nat,nat_of_num,N))) ) ).

% less_num_def
tff(fact_7847_nat__of__num__code_I2_J,axiom,
    ! [N: num] : aa(num,nat,nat_of_num,aa(num,num,bit0,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,N)),aa(num,nat,nat_of_num,N)) ).

% nat_of_num_code(2)
tff(fact_7848_nat__of__num_Osimps_I2_J,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,aa(num,num,bit0,X)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,X)) ).

% nat_of_num.simps(2)
tff(fact_7849_nat__of__num__add,axiom,
    ! [X: num,Y2: num] : aa(num,nat,nat_of_num,aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,Y2)) ).

% nat_of_num_add
tff(fact_7850_nat__of__num__mult,axiom,
    ! [X: num,Y2: num] : aa(num,nat,nat_of_num,aa(num,num,aa(num,fun(num,num),times_times(num),X),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,Y2)) ).

% nat_of_num_mult
tff(fact_7851_nat__of__num__sqr,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,sqr(X)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,X)) ).

% nat_of_num_sqr
tff(fact_7852_nat__of__num_Osimps_I1_J,axiom,
    aa(num,nat,nat_of_num,one2) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_of_num.simps(1)
tff(fact_7853_nat__of__num_Osimps_I3_J,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,aa(num,num,bit1,X)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,X))) ).

% nat_of_num.simps(3)
tff(fact_7854_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(num,nat,nat_of_num,num_of_nat(N)) = N ) ) ).

% num_of_nat_inverse
tff(fact_7855_nat__of__num__code_I3_J,axiom,
    ! [N: num] : aa(num,nat,nat_of_num,aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,N)),aa(num,nat,nat_of_num,N))) ).

% nat_of_num_code(3)
tff(fact_7856_plus__num__def,axiom,
    ! [M2: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N) = num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,M2)),aa(num,nat,nat_of_num,N))) ).

% plus_num_def
tff(fact_7857_arg__max__nat__le,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,F2: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P,X))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,Y3)),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,X)),aa(A,nat,F2,lattices_ord_arg_max(A,nat,F2,P)))) ) ) ).

% arg_max_nat_le
tff(fact_7858_real__floor__code,axiom,
    ! [X: rat] : archim6421214686448440834_floor(real,aa(rat,real,ratreal,X)) = archim6421214686448440834_floor(rat,X) ).

% real_floor_code
tff(fact_7859_prod_H__def,axiom,
    ! [C: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ( groups1962203154675924110t_prod(C,A) = groups_comm_monoid_G(A,C,times_times(A),one_one(A)) ) ) ).

% prod'_def
tff(fact_7860_real__minus__code,axiom,
    ! [X: rat,Y2: rat] : aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(rat,real,ratreal,X)),aa(rat,real,ratreal,Y2)) = aa(rat,real,ratreal,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),X),Y2)) ).

% real_minus_code
tff(fact_7861_real__inverse__code,axiom,
    ! [X: rat] : aa(real,real,inverse_inverse(real),aa(rat,real,ratreal,X)) = aa(rat,real,ratreal,aa(rat,rat,inverse_inverse(rat),X)) ).

% real_inverse_code
tff(fact_7862_real__plus__code,axiom,
    ! [X: rat,Y2: rat] : aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(rat,real,ratreal,X)),aa(rat,real,ratreal,Y2)) = aa(rat,real,ratreal,aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),X),Y2)) ).

% real_plus_code
tff(fact_7863_real__times__code,axiom,
    ! [X: rat,Y2: rat] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(rat,real,ratreal,X)),aa(rat,real,ratreal,Y2)) = aa(rat,real,ratreal,aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),X),Y2)) ).

% real_times_code
tff(fact_7864_one__real__code,axiom,
    one_one(real) = aa(rat,real,ratreal,one_one(rat)) ).

% one_real_code
tff(fact_7865_real__uminus__code,axiom,
    ! [X: rat] : aa(real,real,uminus_uminus(real),aa(rat,real,ratreal,X)) = aa(rat,real,ratreal,aa(rat,rat,uminus_uminus(rat),X)) ).

% real_uminus_code
tff(fact_7866_Ratreal__def,axiom,
    ratreal = field_char_0_of_rat(real) ).

% Ratreal_def
tff(fact_7867_real__less__code,axiom,
    ! [X: rat,Y2: rat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(rat,real,ratreal,X)),aa(rat,real,ratreal,Y2)))
    <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y2)) ) ).

% real_less_code
tff(fact_7868_zero__real__code,axiom,
    zero_zero(real) = aa(rat,real,ratreal,zero_zero(rat)) ).

% zero_real_code
tff(fact_7869_real__less__eq__code,axiom,
    ! [X: rat,Y2: rat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(rat,real,ratreal,X)),aa(rat,real,ratreal,Y2)))
    <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),X),Y2)) ) ).

% real_less_eq_code
tff(fact_7870_real__divide__code,axiom,
    ! [X: rat,Y2: rat] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(rat,real,ratreal,X)),aa(rat,real,ratreal,Y2)) = aa(rat,real,ratreal,aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),X),Y2)) ).

% real_divide_code
tff(fact_7871_sum_H__def,axiom,
    ! [C: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ( groups1027152243600224163dd_sum(C,A) = groups_comm_monoid_G(A,C,plus_plus(A),zero_zero(A)) ) ) ).

% sum'_def
tff(fact_7872_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A3: nat,B2: 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))),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)),product_Pair(nat,product_prod(nat,A),A3),aa(A,product_prod(nat,A),product_Pair(nat,A,B2),Acc))))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc) = Acc ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc) = 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),Acc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_7873_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb3: nat,Xc: A,Y2: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb3,Xc) = Y2 )
     => ( 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))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xb3),Xc))))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb3),Xa))
               => ( Y2 = Xc ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb3),Xa))
               => ( Y2 = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb3,aa(A,A,aa(nat,fun(A,A),X,Xa),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))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xb3),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_7874_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,bool))))] :
      ( 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))),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)),product_Pair(nat,product_prod(nat,A),A1),aa(A,product_prod(nat,A),product_Pair(nat,A,A22),A32))))
     => ( ! [F3: fun(nat,fun(A,A)),A6: nat,B4: 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))),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)),product_Pair(nat,product_prod(nat,A),A6),aa(A,product_prod(nat,A),product_Pair(nat,A,B4),Acc2))))
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B4),A6))
               => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A6),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F3,A6),Acc2))) )
             => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F3),A6),B4),Acc2)) ) )
       => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,A0),A1),A22),A32)) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_7875_prod_Ocomm__monoid__list__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups4802862169904069756st_set(A,times_times(A),one_one(A)) ) ).

% prod.comm_monoid_list_set_axioms
tff(fact_7876_sum_Ocomm__monoid__list__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups4802862169904069756st_set(A,plus_plus(A),zero_zero(A)) ) ).

% sum.comm_monoid_list_set_axioms
tff(fact_7877_natLeq__on__wo__rel,axiom,
    ! [N: nat] : bNF_Wellorder_wo_rel(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_agm(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_wo_rel
tff(fact_7878_max_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice_order(A,ord_max(A),aTP_Lamp_aew(A,fun(A,bool)),aTP_Lamp_aex(A,fun(A,bool))) ) ).

% max.semilattice_order_axioms
tff(fact_7879_semilattice__neutr__order__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
    <=> ( semilattice_neutr(A,F2,Z)
        & semilattice_order(A,F2,Less_eq,Less) ) ) ).

% semilattice_neutr_order_def
tff(fact_7880_semilattice__neutr__order_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( semilattice_neutr(A,F2,Z)
     => ( semilattice_order(A,F2,Less_eq,Less)
       => semila1105856199041335345_order(A,F2,Z,Less_eq,Less) ) ) ).

% semilattice_neutr_order.intro
tff(fact_7881_semilattice__order_Omono,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C2: A,B2: A,D2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2))
       => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),D2))
         => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),aa(A,A,aa(A,fun(A,A),F2,C2),D2))) ) ) ) ).

% semilattice_order.mono
tff(fact_7882_semilattice__order_OorderE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B2) ) ) ) ).

% semilattice_order.orderE
tff(fact_7883_semilattice__order_OorderI,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B2) )
       => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2)) ) ) ).

% semilattice_order.orderI
tff(fact_7884_semilattice__order_Oabsorb1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = A3 ) ) ) ).

% semilattice_order.absorb1
tff(fact_7885_semilattice__order_Oabsorb2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
       => ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = B2 ) ) ) ).

% semilattice_order.absorb2
tff(fact_7886_semilattice__order_Oabsorb3,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = A3 ) ) ) ).

% semilattice_order.absorb3
tff(fact_7887_semilattice__order_Oabsorb4,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),A3))
       => ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = B2 ) ) ) ).

% semilattice_order.absorb4
tff(fact_7888_semilattice__order_OboundedE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)))
       => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2)) ) ) ) ).

% semilattice_order.boundedE
tff(fact_7889_semilattice__order_OboundedI,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2))) ) ) ) ).

% semilattice_order.boundedI
tff(fact_7890_semilattice__order_Oorder__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
      <=> ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B2) ) ) ) ).

% semilattice_order.order_iff
tff(fact_7891_semilattice__order_Ocobounded1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),A3)) ) ).

% semilattice_order.cobounded1
tff(fact_7892_semilattice__order_Ocobounded2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),B2)) ) ).

% semilattice_order.cobounded2
tff(fact_7893_semilattice__order_Oabsorb__iff1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
      <=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = A3 ) ) ) ).

% semilattice_order.absorb_iff1
tff(fact_7894_semilattice__order_Oabsorb__iff2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
      <=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = B2 ) ) ) ).

% semilattice_order.absorb_iff2
tff(fact_7895_semilattice__order_Obounded__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)))
      <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
          & pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2)) ) ) ) ).

% semilattice_order.bounded_iff
tff(fact_7896_semilattice__order_OcoboundedI1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2))
       => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2)) ) ) ).

% semilattice_order.coboundedI1
tff(fact_7897_semilattice__order_OcoboundedI2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,C2: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C2))
       => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2)) ) ) ).

% semilattice_order.coboundedI2
tff(fact_7898_semilattice__order_Ostrict__boundedE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)))
       => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C2)) ) ) ) ).

% semilattice_order.strict_boundedE
tff(fact_7899_semilattice__order_Ostrict__order__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
      <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B2) )
          & ( A3 != B2 ) ) ) ) ).

% semilattice_order.strict_order_iff
tff(fact_7900_semilattice__order_Ostrict__coboundedI1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C2))
       => pp(aa(A,bool,aa(A,fun(A,bool),Less,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2)) ) ) ).

% semilattice_order.strict_coboundedI1
tff(fact_7901_semilattice__order_Ostrict__coboundedI2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,C2: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C2))
       => pp(aa(A,bool,aa(A,fun(A,bool),Less,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2)) ) ) ).

% semilattice_order.strict_coboundedI2
tff(fact_7902_semilattice__neutr__order_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => semilattice_order(A,F2,Less_eq,Less) ) ).

% semilattice_neutr_order.axioms(2)
tff(fact_7903_inf_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => semilattice_order(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).

% inf.semilattice_order_axioms
tff(fact_7904_min_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice_order(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).

% min.semilattice_order_axioms
tff(fact_7905_sup_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semilattice_order(A,sup_sup(A),aTP_Lamp_aey(A,fun(A,bool)),aTP_Lamp_aez(A,fun(A,bool))) ) ).

% sup.semilattice_order_axioms
tff(fact_7906_lexordp__eq_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( ord_lexordp_eq(A,A1,A22)
        <=> ( ? [Ys3: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = Ys3 ) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys3: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4)) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys3: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X4))
                & ord_lexordp_eq(A,Xs3,Ys3) ) ) ) ) ).

% lexordp_eq.simps
tff(fact_7907_lexordp__eq__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A),Y2: A,Ys2: list(A)] :
          ( ord_lexordp_eq(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
              & ord_lexordp_eq(A,Xs,Ys2) ) ) ) ) ).

% lexordp_eq_simps(4)
tff(fact_7908_lexordp__eq_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y2: A,Xs: list(A),Ys2: list(A)] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X))
           => ( ord_lexordp_eq(A,Xs,Ys2)
             => ord_lexordp_eq(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)) ) ) ) ) ).

% lexordp_eq.Cons_eq
tff(fact_7909_lexordp__eq_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y2: A,Xs: list(A),Ys2: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y2))
         => ord_lexordp_eq(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)) ) ) ).

% lexordp_eq.Cons
tff(fact_7910_lexordp__eq_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( ord_lexordp_eq(A,A1,A22)
         => ( ( A1 != nil(A) )
           => ( ! [X2: A] :
                  ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)
                 => ! [Y3: A] :
                      ( ? [Ys5: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3)) ) )
             => ~ ! [X2: A,Y3: A,Xs2: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
                   => ! [Ys5: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
                       => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y3))
                         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X2))
                           => ~ ord_lexordp_eq(A,Xs2,Ys5) ) ) ) ) ) ) ) ) ).

% lexordp_eq.cases
tff(fact_7911_those_Osimps_I2_J,axiom,
    ! [A: $tType,X: option(A),Xs: list(option(A))] : those(A,aa(list(option(A)),list(option(A)),aa(option(A),fun(list(option(A)),list(option(A))),cons(option(A)),X),Xs)) = aa(option(A),option(list(A)),aa(fun(A,option(list(A))),fun(option(A),option(list(A))),aa(option(list(A)),fun(fun(A,option(list(A))),fun(option(A),option(list(A)))),case_option(option(list(A)),A),none(list(A))),aTP_Lamp_ahm(list(option(A)),fun(A,option(list(A))),Xs)),X) ).

% those.simps(2)
tff(fact_7912_natLeq__natLess__Id,axiom,
    bNF_Ca8459412986667044542atLess = 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))),minus_minus(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),id2(nat)) ).

% natLeq_natLess_Id
tff(fact_7913_relpow_Osimps_I1_J,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))),zero_zero(nat)),R2) = id2(A) ).

% relpow.simps(1)
tff(fact_7914_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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)))
       => ( ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),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))))
              | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),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)))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
         => ( ( ( A3 = B2 )
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_7915_ntrancl__Suc,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,aa(nat,nat,suc,N),R2) = relcomp(A,A,A,transitive_ntrancl(A,N,R2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),id2(A)),R2)) ).

% ntrancl_Suc
tff(fact_7916_relpow_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),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))),N),R2),R2) ).

% relpow.simps(2)
tff(fact_7917_Field__natLeq__on,axiom,
    ! [N: nat] : field2(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_agm(nat,fun(nat,fun(nat,bool)),N)))) = collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N)) ).

% Field_natLeq_on
tff(fact_7918_bsqr__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R) = collect(product_prod(product_prod(A,A),product_prod(A,A)),product_case_prod(product_prod(A,A),product_prod(A,A),bool,product_case_prod(A,A,fun(product_prod(A,A),bool),aTP_Lamp_aho(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),R)))) ).

% bsqr_def
tff(fact_7919_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)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A3),B2)),R))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),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_7920_strict__linear__order__on__diff__Id,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,A4,R)
     => order_5396836661320670305der_on(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))),minus_minus(set(product_prod(A,A))),R),id2(A))) ) ).

% strict_linear_order_on_diff_Id
tff(fact_7921_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)))
      <=> ! [A14: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A14),field2(A,R)))
           => ( ( A14 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A14))
                  & ! [Xa3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A14))
                     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X4),Xa3)),R)) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_7922_wf,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wf(A,collect(product_prod(A,A),product_case_prod(A,A,bool,ord_less(A)))) ) ).

% wf
tff(fact_7923_wf__if__measure,axiom,
    ! [A: $tType,P: fun(A,bool),F2: fun(A,nat),G: fun(A,A)] :
      ( ! [X2: A] :
          ( pp(aa(A,bool,P,X2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,aa(A,A,G,X2))),aa(A,nat,F2,X2))) )
     => wf(A,collect(product_prod(A,A),product_case_prod(A,A,bool,aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_ahp(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),P),G)))) ) ).

% wf_if_measure
tff(fact_7924_wf__less,axiom,
    wf(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,ord_less(nat)))) ).

% wf_less
tff(fact_7925_wf__bounded__measure,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Ub: fun(A,nat),F2: fun(A,nat)] :
      ( ! [A6: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,B4),A6)),R))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Ub,B4)),aa(A,nat,Ub,A6)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,B4)),aa(A,nat,Ub,A6)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,A6)),aa(A,nat,F2,B4))) ) )
     => wf(A,R) ) ).

% wf_bounded_measure
tff(fact_7926_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ~ ? [F5: fun(nat,A)] :
          ! [I2: nat] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F5,aa(nat,nat,suc,I2))),aa(nat,A,F5,I2))),R)) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_7927_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),F2: fun(nat,A)] :
      ( wf(A,R)
     => ~ ! [K2: nat] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F2,aa(nat,nat,suc,K2))),aa(nat,A,F2,K2))),R)) ) ).

% wf_no_infinite_down_chainE
tff(fact_7928_wf__int__ge__less__than,axiom,
    ! [D2: int] : wf(int,int_ge_less_than(D2)) ).

% wf_int_ge_less_than
tff(fact_7929_wf__int__ge__less__than2,axiom,
    ! [D2: int] : wf(int,int_ge_less_than2(D2)) ).

% wf_int_ge_less_than2
tff(fact_7930_wo__rel_OWF,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( bNF_Wellorder_wo_rel(A,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))) ) ).

% wo_rel.WF
tff(fact_7931_partial__order__on__well__order__on,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( order_7125193373082350890der_on(A,A4,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))) ) ) ).

% partial_order_on_well_order_on
tff(fact_7932_dir__image__minus__Id,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),R: set(product_prod(A,A))] :
      ( inj_on(A,B,F2,field2(A,R))
     => ( aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),minus_minus(set(product_prod(B,B))),bNF_We2720479622203943262_image(A,B,R,F2)),id2(B)) = bNF_We2720479622203943262_image(A,B,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)),F2) ) ) ).

% dir_image_minus_Id
tff(fact_7933_Total__Id__Field,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R),R)
     => ( ~ pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),id2(A)))
       => ( field2(A,R) = field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A))) ) ) ) ).

% Total_Id_Field
tff(fact_7934_partial__order__on__acyclic,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,A4,R)
     => transitive_acyclic(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))) ) ).

% partial_order_on_acyclic
tff(fact_7935_total__on__diff__Id,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( total_on(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))),minus_minus(set(product_prod(A,A))),R),id2(A)))
    <=> total_on(A,A4,R) ) ).

% total_on_diff_Id
tff(fact_7936_acyclicI__order,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [R: set(product_prod(B,B)),F2: fun(B,A)] :
          ( ! [A6: B,B4: B] :
              ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),product_Pair(B,B,A6),B4)),R))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,B4)),aa(B,A,F2,A6))) )
         => transitive_acyclic(B,R) ) ) ).

% acyclicI_order
tff(fact_7937_linear__order__on__acyclic,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,A4,R)
     => transitive_acyclic(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))) ) ).

% linear_order_on_acyclic
tff(fact_7938_natLeq__on__Well__order,axiom,
    ! [N: nat] : order_well_order_on(nat,field2(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_agm(nat,fun(nat,fun(nat,bool)),N)))),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_agm(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_Well_order
tff(fact_7939_well__order__on__def,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( order_well_order_on(A,A4,R)
    <=> ( order_679001287576687338der_on(A,A4,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))) ) ) ).

% well_order_on_def
tff(fact_7940_natLeq__on__well__order__on,axiom,
    ! [N: nat] : order_well_order_on(nat,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),N)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_agm(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_well_order_on
tff(fact_7941_irrefl__diff__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : irrefl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A))) ).

% irrefl_diff_Id
tff(fact_7942_Range__Diff__subset,axiom,
    ! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B5: set(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),range2(B,A,A4)),range2(B,A,B5))),range2(B,A,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))),minus_minus(set(product_prod(B,A))),A4),B5)))) ).

% Range_Diff_subset
tff(fact_7943_bdd__below_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,aTP_Lamp_ahq(A,fun(A,bool)),aTP_Lamp_aeo(A,fun(A,bool))) ) ).

% bdd_below.preordering_bdd_axioms
tff(fact_7944_tendsto__Zfun__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A3: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F4)
        <=> zfun(A,B,aa(B,fun(A,B),aTP_Lamp_tr(fun(A,B),fun(B,fun(A,B)),F2),A3),F4) ) ) ).

% tendsto_Zfun_iff
tff(fact_7945_bdd__above_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,ord_less_eq(A),ord_less(A)) ) ).

% bdd_above.preordering_bdd_axioms
tff(fact_7946_Zfun__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F4: filter(A)] : zfun(A,B,aTP_Lamp_ahr(A,B),F4) ) ).

% Zfun_zero
tff(fact_7947_Zfun__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( zfun(A,B,F2,F4)
         => zfun(A,B,aTP_Lamp_ahs(fun(A,B),fun(A,B),F2),F4) ) ) ).

% Zfun_minus
tff(fact_7948_Zfun__diff,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_aht(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% Zfun_diff
tff(fact_7949_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_zero(code_natural) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,N)),aa(nat,nat,suc,zero_zero(nat)))),aa(code_natural,nat,code_nat_of_natural,N))) ) ).

% natural_decr
tff(fact_7950_Domain__Diff__subset,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B5: set(product_prod(A,B))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),domain(A,B,A4)),domain(A,B,B5))),domain(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),A4),B5)))) ).

% Domain_Diff_subset
tff(fact_7951_minus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).

% minus_natural.rep_eq
tff(fact_7952_one__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,one_one(code_natural)) = one_one(nat) ).

% one_natural.rep_eq
tff(fact_7953_divide__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),X),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).

% divide_natural.rep_eq
tff(fact_7954_zero__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,zero_zero(code_natural)) = zero_zero(nat) ).

% zero_natural.rep_eq
tff(fact_7955_natural__zero__minus__one,axiom,
    aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),zero_zero(code_natural)),one_one(code_natural)) = zero_zero(code_natural) ).

% natural_zero_minus_one
tff(fact_7956_less__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),X),Xa))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa))) ) ).

% less_natural.rep_eq
tff(fact_7957_int__of__integer__of__natural,axiom,
    ! [N: code_natural] : aa(code_integer,int,code_int_of_integer,aa(code_natural,code_integer,code_i5400310926305786745atural,N)) = aa(nat,int,semiring_1_of_nat(int),aa(code_natural,nat,code_nat_of_natural,N)) ).

% int_of_integer_of_natural
tff(fact_7958_integer__of__natural_Orep__eq,axiom,
    ! [X: code_natural] : aa(code_integer,int,code_int_of_integer,aa(code_natural,code_integer,code_i5400310926305786745atural,X)) = aa(nat,int,semiring_1_of_nat(int),aa(code_natural,nat,code_nat_of_natural,X)) ).

% integer_of_natural.rep_eq
tff(fact_7959_log_Osimps,axiom,
    ! [B2: code_natural,I: code_natural] :
      ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
          | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I),B2)) )
       => ( log(B2,I) = one_one(code_natural) ) )
      & ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
            | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I),B2)) )
       => ( log(B2,I) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(B2,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),I),B2))) ) ) ) ).

% log.simps
tff(fact_7960_log_Oelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y2: code_natural] :
      ( ( log(X,Xa) = Y2 )
     => ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
            | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X)) )
         => ( Y2 = one_one(code_natural) ) )
        & ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
              | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X)) )
         => ( Y2 = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa),X))) ) ) ) ) ).

% log.elims
tff(fact_7961_minus__shift__def,axiom,
    ! [K: code_natural,L: code_natural,R: code_natural] :
      ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),K),L))
       => ( minus_shift(R,K,L) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),R),K)),L) ) )
      & ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),K),L))
       => ( minus_shift(R,K,L) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),L) ) ) ) ).

% minus_shift_def
tff(fact_7962_next_Osimps,axiom,
    ! [V2: code_natural,W: code_natural] : aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,V2),W)) = aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,product_prod(code_natural,code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),V2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2)))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,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,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),W),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))),one_one(code_natural)))),one_one(code_natural))),aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),V2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,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,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),W),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))) ).

% next.simps
tff(fact_7963_Random_Orange__def,axiom,
    ! [K: code_natural] : range(K) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),iterate(code_natural,product_prod(code_natural,code_natural),log(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),K),aTP_Lamp_ahv(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))))),one_one(code_natural)),aTP_Lamp_ahw(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K)) ).

% Random.range_def
tff(fact_7964_inc__shift__def,axiom,
    ! [V2: code_natural,K: code_natural] :
      ( ( ( V2 = K )
       => ( inc_shift(V2,K) = one_one(code_natural) ) )
      & ( ( V2 != K )
       => ( inc_shift(V2,K) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),K),one_one(code_natural)) ) ) ) ).

% inc_shift_def
tff(fact_7965_iterate_Oelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb3: B,Y2: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb3) = Y2 )
     => ( ( ( X = zero_zero(code_natural) )
         => ( Y2 = product_Pair(B,A,Xb3) ) )
        & ( ( X != zero_zero(code_natural) )
         => ( Y2 = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb3),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa)) ) ) ) ) ).

% iterate.elims
tff(fact_7966_iterate_Osimps,axiom,
    ! [A: $tType,B: $tType,K: code_natural,F2: fun(B,fun(A,product_prod(B,A))),X: B] :
      ( ( ( K = zero_zero(code_natural) )
       => ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F2),X) = product_Pair(B,A,X) ) )
      & ( ( K != zero_zero(code_natural) )
       => ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F2),X) = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F2,X),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),one_one(code_natural)),F2)) ) ) ) ).

% iterate.simps
tff(fact_7967_log_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y2: code_natural] :
      ( ( log(X,Xa) = Y2 )
     => ( accp(product_prod(code_natural,code_natural),log_rel,aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,X),Xa))
       => ~ ( ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
                  | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X)) )
               => ( Y2 = one_one(code_natural) ) )
              & ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
                    | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X)) )
               => ( Y2 = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa),X))) ) ) )
           => ~ accp(product_prod(code_natural,code_natural),log_rel,aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,X),Xa)) ) ) ) ).

% log.pelims
tff(fact_7968_iterate_Opelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb3: B,Y2: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb3) = Y2 )
     => ( accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),product_Pair(fun(B,fun(A,product_prod(B,A))),B,Xa),Xb3)))
       => ~ ( ( ( ( X = zero_zero(code_natural) )
               => ( Y2 = product_Pair(B,A,Xb3) ) )
              & ( ( X != zero_zero(code_natural) )
               => ( Y2 = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb3),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa)) ) ) )
           => ~ accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),product_Pair(fun(B,fun(A,product_prod(B,A))),B,Xa),Xb3))) ) ) ) ).

% iterate.pelims
tff(fact_7969_select__def,axiom,
    ! [A: $tType,Xs: list(A)] : select(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(aa(nat,code_natural,code_natural_of_nat,aa(list(A),nat,size_size(list(A)),Xs))),aTP_Lamp_ahx(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).

% select_def
tff(fact_7970_integer__of__natural__def,axiom,
    code_i5400310926305786745atural = aa(fun(nat,int),fun(code_natural,code_integer),map_fun(code_natural,nat,int,code_integer,code_nat_of_natural,code_integer_of_int),semiring_1_of_nat(int)) ).

% integer_of_natural_def
tff(fact_7971_zero__natural__def,axiom,
    zero_zero(code_natural) = aa(nat,code_natural,code_natural_of_nat,zero_zero(nat)) ).

% zero_natural_def
tff(fact_7972_minus__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),X)) ).

% minus_natural.abs_eq
tff(fact_7973_divide__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),X)) ).

% divide_natural.abs_eq
tff(fact_7974_less__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),X)) ) ).

% less_natural.abs_eq
tff(fact_7975_one__natural__def,axiom,
    one_one(code_natural) = aa(nat,code_natural,code_natural_of_nat,one_one(nat)) ).

% one_natural_def
tff(fact_7976_integer__of__natural_Oabs__eq,axiom,
    ! [X: nat] : aa(code_natural,code_integer,code_i5400310926305786745atural,aa(nat,code_natural,code_natural_of_nat,X)) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),X)) ).

% integer_of_natural.abs_eq
tff(fact_7977_pick__same,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pick(A,map(A,product_prod(code_natural,A),product_Pair(code_natural,A,one_one(code_natural)),Xs),aa(nat,code_natural,code_natural_of_nat,L)) = aa(nat,A,nth(A,Xs),L) ) ) ).

% pick_same
tff(fact_7978_select__weight__select,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( select_weight(A,map(A,product_prod(code_natural,A),product_Pair(code_natural,A,one_one(code_natural)),Xs)) = select(A,Xs) ) ) ).

% select_weight_select
tff(fact_7979_pick_Osimps,axiom,
    ! [A: $tType,I: code_natural,X: product_prod(code_natural,A),Xs: list(product_prod(code_natural,A))] :
      ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X)))
       => ( pick(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),aa(product_prod(code_natural,A),fun(list(product_prod(code_natural,A)),list(product_prod(code_natural,A))),cons(product_prod(code_natural,A)),X),Xs),I) = aa(product_prod(code_natural,A),A,product_snd(code_natural,A),X) ) )
      & ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X)))
       => ( pick(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),aa(product_prod(code_natural,A),fun(list(product_prod(code_natural,A)),list(product_prod(code_natural,A))),cons(product_prod(code_natural,A)),X),Xs),I) = pick(A,Xs,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),I),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X))) ) ) ) ).

% pick.simps
tff(fact_7980_Suc_Orep__eq,axiom,
    ! [X: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,code_Suc,X)) = aa(nat,nat,suc,aa(code_natural,nat,code_nat_of_natural,X)) ).

% Suc.rep_eq
tff(fact_7981_Suc__natural__minus__one,axiom,
    ! [N: code_natural] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(code_natural,code_natural,code_Suc,N)),one_one(code_natural)) = N ).

% Suc_natural_minus_one
tff(fact_7982_Suc_Oabs__eq,axiom,
    ! [X: nat] : aa(code_natural,code_natural,code_Suc,aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,suc,X)) ).

% Suc.abs_eq
tff(fact_7983_Code__Numeral_OSuc__def,axiom,
    code_Suc = aa(fun(nat,nat),fun(code_natural,code_natural),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat),suc) ).

% Code_Numeral.Suc_def
tff(fact_7984_semilattice__set_Oeq__fold_H,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( lattic1715443433743089157tice_F(A,F2,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_ahy(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),F2),none(A),A4)) ) ) ).

% semilattice_set.eq_fold'
tff(fact_7985_semilattice__set_Oinsert__remove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),X: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( pp(aa(set(A),bool,finite_finite(A),A4))
       => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
           => ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),insert(A,X),A4)) = X ) )
          & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
           => ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),insert(A,X),A4)) = aa(A,A,aa(A,fun(A,A),F2,X),lattic1715443433743089157tice_F(A,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,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% semilattice_set.insert_remove
tff(fact_7986_semilattice__set_Oremove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),X: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( pp(aa(set(A),bool,finite_finite(A),A4))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A4))
         => ( ( ( 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,X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( lattic1715443433743089157tice_F(A,F2,A4) = X ) )
            & ( ( 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,X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( lattic1715443433743089157tice_F(A,F2,A4) = aa(A,A,aa(A,fun(A,A),F2,X),lattic1715443433743089157tice_F(A,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,X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% semilattice_set.remove
tff(fact_7987_semilattice__set_Oinfinite,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( ~ pp(aa(set(A),bool,finite_finite(A),A4))
       => ( lattic1715443433743089157tice_F(A,F2,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% semilattice_set.infinite
tff(fact_7988_divide__natural__def,axiom,
    divide_divide(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),divide_divide(nat)) ).

% divide_natural_def
tff(fact_7989_minus__natural__def,axiom,
    minus_minus(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),minus_minus(nat)) ).

% minus_natural_def
tff(fact_7990_Quotient3__int,axiom,
    quotient3(product_prod(nat,nat),int,intrel,abs_Integ,rep_Integ) ).

% Quotient3_int
tff(fact_7991_map__option__o__case__sum,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F2: fun(D,C),G: fun(A,option(D)),H: fun(B,option(D))] : aa(fun(sum_sum(A,B),option(D)),fun(sum_sum(A,B),option(C)),comp(option(D),option(C),sum_sum(A,B),aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F2)),sum_case_sum(A,option(D),B,G,H)) = sum_case_sum(A,option(C),B,aa(fun(A,option(D)),fun(A,option(C)),comp(option(D),option(C),A,aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F2)),G),aa(fun(B,option(D)),fun(B,option(C)),comp(option(D),option(C),B,aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F2)),H)) ).

% map_option_o_case_sum
tff(fact_7992_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y2: bool] :
      ( ( vEBT_VEBT_minNull(X)
      <=> pp(Y2) )
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y2)
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fFalse,fFalse)) ) )
         => ( ! [Uv2: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv2) )
               => ( ~ pp(Y2)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fTrue,Uv2)) ) )
           => ( ! [Uu2: bool] :
                  ( ( X = vEBT_Leaf(Uu2,fTrue) )
                 => ( ~ pp(Y2)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(Uu2,fTrue)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy) )
                   => ( pp(Y2)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ~ pp(Y2)
                       => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_7993_is__none__bind,axiom,
    ! [A: $tType,B: $tType,F2: option(B),G: fun(B,option(A))] :
      ( is_none(A,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),F2),G))
    <=> ( is_none(B,F2)
        | is_none(A,aa(B,option(A),G,aa(option(B),B,the2(B),F2))) ) ) ).

% is_none_bind
tff(fact_7994_is__none__code_I2_J,axiom,
    ! [B: $tType,X: B] : ~ is_none(B,aa(B,option(B),some(B),X)) ).

% is_none_code(2)
tff(fact_7995_is__none__code_I1_J,axiom,
    ! [A: $tType] : is_none(A,none(A)) ).

% is_none_code(1)
tff(fact_7996_is__none__map__option,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: option(B)] :
      ( is_none(A,aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F2),X))
    <=> is_none(B,X) ) ).

% is_none_map_option
tff(fact_7997_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(X)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fFalse,fFalse)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_7998_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(X)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ! [Uv2: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv2) )
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fTrue,Uv2)) )
         => ( ! [Uu2: bool] :
                ( ( X = vEBT_Leaf(Uu2,fTrue) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(Uu2,fTrue)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,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),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_7999_is__none__simps_I2_J,axiom,
    ! [B: $tType,X: B] : ~ is_none(B,aa(B,option(B),some(B),X)) ).

% is_none_simps(2)
tff(fact_8000_Option_Ois__none__def,axiom,
    ! [A: $tType,X: option(A)] :
      ( is_none(A,X)
    <=> ( X = none(A) ) ) ).

% Option.is_none_def
tff(fact_8001_is__none__simps_I1_J,axiom,
    ! [A: $tType] : is_none(A,none(A)) ).

% is_none_simps(1)
tff(fact_8002_the__map__option,axiom,
    ! [B: $tType,A: $tType,X: option(A),F2: fun(A,B)] :
      ( ~ is_none(A,X)
     => ( 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),X)) = aa(A,B,F2,aa(option(A),A,the2(A),X)) ) ) ).

% the_map_option
tff(fact_8003_rel__option__unfold,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: option(A),Y2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y2))
    <=> ( ( is_none(A,X)
        <=> is_none(B,Y2) )
        & ( ~ is_none(A,X)
         => ( ~ is_none(B,Y2)
           => pp(aa(B,bool,aa(A,fun(B,bool),R2,aa(option(A),A,the2(A),X)),aa(option(B),B,the2(B),Y2))) ) ) ) ) ).

% rel_option_unfold
tff(fact_8004_rel__optionI,axiom,
    ! [A: $tType,B: $tType,X: option(A),Y2: option(B),P: fun(A,fun(B,bool))] :
      ( ( is_none(A,X)
      <=> is_none(B,Y2) )
     => ( ( ~ is_none(A,X)
         => ( ~ is_none(B,Y2)
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(option(A),A,the2(A),X)),aa(option(B),B,the2(B),Y2))) ) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P),X),Y2)) ) ) ).

% rel_optionI
tff(fact_8005_sub_Otransfer,axiom,
    pp(aa(fun(num,fun(num,code_integer)),bool,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,code_integer)),bool),bNF_rel_fun(num,num,fun(num,int),fun(num,code_integer),fequal(num),bNF_rel_fun(num,num,int,code_integer,fequal(num),code_pcr_integer)),aTP_Lamp_ot(num,fun(num,int))),code_sub)) ).

% sub.transfer
tff(fact_8006_bounded__bilinear_Otendsto__right__zero,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,B),F4: filter(D),C2: A] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_ahz(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Prod),F2),C2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% bounded_bilinear.tendsto_right_zero
tff(fact_8007_one__integer_Otransfer,axiom,
    pp(aa(code_integer,bool,aa(int,fun(code_integer,bool),code_pcr_integer,one_one(int)),one_one(code_integer))) ).

% one_integer.transfer
tff(fact_8008_less__integer_Otransfer,axiom,
    pp(aa(fun(code_integer,fun(code_integer,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(code_integer,fun(code_integer,bool)),bool),bNF_rel_fun(int,code_integer,fun(int,bool),fun(code_integer,bool),code_pcr_integer,bNF_rel_fun(int,code_integer,bool,bool,code_pcr_integer,fequal(bool))),ord_less(int)),ord_less(code_integer))) ).

% less_integer.transfer
tff(fact_8009_divide__integer_Otransfer,axiom,
    pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),divide_divide(int)),divide_divide(code_integer))) ).

% divide_integer.transfer
tff(fact_8010_minus__integer_Otransfer,axiom,
    pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),minus_minus(int)),minus_minus(code_integer))) ).

% minus_integer.transfer
tff(fact_8011_bounded__bilinear_Odiff__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),minus_minus(B),B2),B6)) = aa(C,C,aa(C,fun(C,C),minus_minus(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.diff_right
tff(fact_8012_bounded__bilinear_Odiff__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),minus_minus(A),A3),A7)),B2) = aa(C,C,aa(C,fun(C,C),minus_minus(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.diff_left
tff(fact_8013_uminus__integer_Otransfer,axiom,
    pp(aa(fun(code_integer,code_integer),bool,aa(fun(int,int),fun(fun(code_integer,code_integer),bool),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),uminus_uminus(int)),uminus_uminus(code_integer))) ).

% uminus_integer.transfer
tff(fact_8014_integer__of__nat_Otransfer,axiom,
    pp(aa(fun(nat,code_integer),bool,aa(fun(nat,int),fun(fun(nat,code_integer),bool),bNF_rel_fun(nat,nat,int,code_integer,fequal(nat),code_pcr_integer),semiring_1_of_nat(int)),code_integer_of_nat)) ).

% integer_of_nat.transfer
tff(fact_8015_bounded__bilinear_Ominus__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] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,A3),aa(B,B,uminus_uminus(B),B2)) = aa(C,C,uminus_uminus(C),aa(B,C,aa(A,fun(B,C),Prod,A3),B2)) ) ) ) ).

% bounded_bilinear.minus_right
tff(fact_8016_bounded__bilinear_Ominus__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,B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,uminus_uminus(A),A3)),B2) = aa(C,C,uminus_uminus(C),aa(B,C,aa(A,fun(B,C),Prod,A3),B2)) ) ) ) ).

% bounded_bilinear.minus_left
tff(fact_8017_bounded__bilinear_Ozero__left,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,zero_zero(A)),B2) = zero_zero(C) ) ) ) ).

% bounded_bilinear.zero_left
tff(fact_8018_bounded__bilinear_Ozero__right,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),A3: A] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,A3),zero_zero(B)) = zero_zero(C) ) ) ) ).

% bounded_bilinear.zero_right
tff(fact_8019_zero__integer_Otransfer,axiom,
    pp(aa(code_integer,bool,aa(int,fun(code_integer,bool),code_pcr_integer,zero_zero(int)),zero_zero(code_integer))) ).

% zero_integer.transfer
tff(fact_8020_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)),X: A,Y2: 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,X),Y2)),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),X),A3)),aa(B,B,aa(B,fun(B,B),minus_minus(B),Y2),B2))),aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3)),B2))),aa(B,C,aa(A,fun(B,C),Prod,A3),aa(B,B,aa(B,fun(B,B),minus_minus(B),Y2),B2))) ) ) ) ).

% bounded_bilinear.prod_diff_prod
tff(fact_8021_bounded__bilinear_Otendsto__zero,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,A),F4: filter(D),G: fun(D,B)] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
             => filterlim(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_aia(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Prod),F2),G),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ) ).

% bounded_bilinear.tendsto_zero
tff(fact_8022_bounded__bilinear_Otendsto__left__zero,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,A),F4: filter(D),C2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(D,C,aa(B,fun(D,C),aa(fun(D,A),fun(B,fun(D,C)),aTP_Lamp_aib(fun(A,fun(B,C)),fun(fun(D,A),fun(B,fun(D,C))),Prod),F2),C2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% bounded_bilinear.tendsto_left_zero
tff(fact_8023_integer__of__natural_Otransfer,axiom,
    pp(aa(fun(code_natural,code_integer),bool,aa(fun(nat,int),fun(fun(code_natural,code_integer),bool),bNF_rel_fun(nat,code_natural,int,code_integer,code_pcr_natural,code_pcr_integer),semiring_1_of_nat(int)),code_i5400310926305786745atural)) ).

% integer_of_natural.transfer
tff(fact_8024_list__all__length,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( list_all(A,P,Xs)
    <=> ! [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N5))) ) ) ).

% list_all_length
tff(fact_8025_Suc_Otransfer,axiom,
    pp(aa(fun(code_natural,code_natural),bool,aa(fun(nat,nat),fun(fun(code_natural,code_natural),bool),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural),suc),code_Suc)) ).

% Suc.transfer
tff(fact_8026_divide__natural_Otransfer,axiom,
    pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),divide_divide(nat)),divide_divide(code_natural))) ).

% divide_natural.transfer
tff(fact_8027_less__natural_Otransfer,axiom,
    pp(aa(fun(code_natural,fun(code_natural,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(code_natural,fun(code_natural,bool)),bool),bNF_rel_fun(nat,code_natural,fun(nat,bool),fun(code_natural,bool),code_pcr_natural,bNF_rel_fun(nat,code_natural,bool,bool,code_pcr_natural,fequal(bool))),ord_less(nat)),ord_less(code_natural))) ).

% less_natural.transfer
tff(fact_8028_minus__natural_Otransfer,axiom,
    pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),minus_minus(nat)),minus_minus(code_natural))) ).

% minus_natural.transfer
tff(fact_8029_zero__natural_Otransfer,axiom,
    pp(aa(code_natural,bool,aa(nat,fun(code_natural,bool),code_pcr_natural,zero_zero(nat)),zero_zero(code_natural))) ).

% zero_natural.transfer
tff(fact_8030_one__natural_Otransfer,axiom,
    pp(aa(code_natural,bool,aa(nat,fun(code_natural,bool),code_pcr_natural,one_one(nat)),one_one(code_natural))) ).

% one_natural.transfer
tff(fact_8031_linear__injective__on__subspace__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( real_Vector_linear(A,B,F2)
         => ( real_Vector_subspace(A,S)
           => ( inj_on(A,B,F2,S)
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
                 => ( ( aa(A,B,F2,X4) = zero_zero(B) )
                   => ( X4 = zero_zero(A) ) ) ) ) ) ) ) ).

% linear_injective_on_subspace_0
tff(fact_8032_of__int__code_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),neg(K)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_code(1)
tff(fact_8033_less__int__code_I3_J,axiom,
    ! [L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),neg(L))) ).

% less_int_code(3)
tff(fact_8034_less__int__code_I7_J,axiom,
    ! [K: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),neg(K)),zero_zero(int))) ).

% less_int_code(7)
tff(fact_8035_less__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),neg(K)),neg(L)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),L),K)) ) ).

% less_int_code(9)
tff(fact_8036_subspace__diff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A),X: A,Y2: A] :
          ( real_Vector_subspace(A,S3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),S3))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y2)),S3)) ) ) ) ) ).

% subspace_diff
tff(fact_8037_subspace__neg,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A),X: A] :
          ( real_Vector_subspace(A,S3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,uminus_uminus(A),X)),S3)) ) ) ) ).

% subspace_neg
tff(fact_8038_subspace__single__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => real_Vector_subspace(A,aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A)))) ) ).

% subspace_single_0
tff(fact_8039_linear__subspace__kernel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => real_Vector_subspace(A,collect(A,aTP_Lamp_aic(fun(A,B),fun(A,bool),F2))) ) ) ).

% linear_subspace_kernel
tff(fact_8040_subspace__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( real_Vector_subspace(A,S3)
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),S3)) ) ) ).

% subspace_0
tff(fact_8041_nat__code_I1_J,axiom,
    ! [K: num] : aa(int,nat,nat2,neg(K)) = zero_zero(nat) ).

% nat_code(1)
tff(fact_8042_subspaceI,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),S3))
         => ( ! [X2: A,Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),S3))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y3)),S3)) ) )
           => ( ! [C3: real,X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S3))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),real_V8093663219630862766scaleR(A,C3,X2)),S3)) )
             => real_Vector_subspace(A,S3) ) ) ) ) ).

% subspaceI
tff(fact_8043_subspace__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( real_Vector_subspace(A,S3)
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),S3))
            & ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
               => ! [Xa3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),S3))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Xa3)),S3)) ) )
            & ! [C4: real,X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
               => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),real_V8093663219630862766scaleR(A,C4,X4)),S3)) ) ) ) ) ).

% subspace_def
tff(fact_8044_less__eq__int__code_I7_J,axiom,
    ! [K: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),neg(K)),zero_zero(int))) ).

% less_eq_int_code(7)
tff(fact_8045_less__eq__int__code_I3_J,axiom,
    ! [L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),neg(L))) ).

% less_eq_int_code(3)
tff(fact_8046_less__eq__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),neg(K)),neg(L)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),L),K)) ) ).

% less_eq_int_code(9)
tff(fact_8047_plus__int__code_I6_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),neg(M2)),neg(N)) = neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)) ).

% plus_int_code(6)
tff(fact_8048_Int_Osub__code_I4_J,axiom,
    ! [N: num] : sub(one2,aa(num,num,bit0,N)) = neg(bitM(N)) ).

% Int.sub_code(4)
tff(fact_8049_Int_Osub__code_I5_J,axiom,
    ! [N: num] : sub(one2,aa(num,num,bit1,N)) = neg(aa(num,num,bit0,N)) ).

% Int.sub_code(5)
tff(fact_8050_minus__int__code_I6_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),neg(M2)),neg(N)) = sub(N,M2) ).

% minus_int_code(6)
tff(fact_8051_Int_Osub__def,axiom,
    ! [M2: num,N: num] : sub(M2,N) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) ).

% Int.sub_def
tff(fact_8052_Int_Osub__code_I1_J,axiom,
    sub(one2,one2) = zero_zero(int) ).

% Int.sub_code(1)
tff(fact_8053_Int_Osub__code_I9_J,axiom,
    ! [M2: num,N: num] : sub(aa(num,num,bit0,M2),aa(num,num,bit1,N)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),dup(sub(M2,N))),one_one(int)) ).

% Int.sub_code(9)
tff(fact_8054_Int_Osub__code_I8_J,axiom,
    ! [M2: num,N: num] : sub(aa(num,num,bit1,M2),aa(num,num,bit0,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),dup(sub(M2,N))),one_one(int)) ).

% Int.sub_code(8)
tff(fact_8055_Int_Odup__code_I1_J,axiom,
    dup(zero_zero(int)) = zero_zero(int) ).

% Int.dup_code(1)
tff(fact_8056_Int_Odup__def,axiom,
    ! [K: int] : dup(K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),K) ).

% Int.dup_def
tff(fact_8057_Int_Odup__code_I3_J,axiom,
    ! [N: num] : dup(neg(N)) = neg(aa(num,num,bit0,N)) ).

% Int.dup_code(3)
tff(fact_8058_Int_Osub__code_I6_J,axiom,
    ! [M2: num,N: num] : sub(aa(num,num,bit0,M2),aa(num,num,bit0,N)) = dup(sub(M2,N)) ).

% Int.sub_code(6)
tff(fact_8059_Int_Osub__code_I7_J,axiom,
    ! [M2: num,N: num] : sub(aa(num,num,bit1,M2),aa(num,num,bit1,N)) = dup(sub(M2,N)) ).

% Int.sub_code(7)
tff(fact_8060_Int_Osub__code_I2_J,axiom,
    ! [M2: num] : sub(aa(num,num,bit0,M2),one2) = aa(num,int,pos,bitM(M2)) ).

% Int.sub_code(2)
tff(fact_8061_Int_Osub__code_I3_J,axiom,
    ! [M2: num] : sub(aa(num,num,bit1,M2),one2) = aa(num,int,pos,aa(num,num,bit0,M2)) ).

% Int.sub_code(3)
tff(fact_8062_Int_Odup__code_I2_J,axiom,
    ! [N: num] : dup(aa(num,int,pos,N)) = aa(num,int,pos,aa(num,num,bit0,N)) ).

% Int.dup_code(2)
tff(fact_8063_less__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,pos,K)),neg(L))) ).

% less_int_code(6)
tff(fact_8064_less__int__code_I8_J,axiom,
    ! [K: num,L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),neg(K)),aa(num,int,pos,L))) ).

% less_int_code(8)
tff(fact_8065_uminus__int__code_I3_J,axiom,
    ! [M2: num] : aa(int,int,uminus_uminus(int),neg(M2)) = aa(num,int,pos,M2) ).

% uminus_int_code(3)
tff(fact_8066_uminus__int__code_I2_J,axiom,
    ! [M2: num] : aa(int,int,uminus_uminus(int),aa(num,int,pos,M2)) = neg(M2) ).

% uminus_int_code(2)
tff(fact_8067_Int_ONeg__def,axiom,
    ! [N: num] : neg(N) = aa(int,int,uminus_uminus(int),aa(num,int,pos,N)) ).

% Int.Neg_def
tff(fact_8068_less__eq__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,pos,K)),neg(L))) ).

% less_eq_int_code(6)
tff(fact_8069_less__eq__int__code_I8_J,axiom,
    ! [K: num,L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),neg(K)),aa(num,int,pos,L))) ).

% less_eq_int_code(8)
tff(fact_8070_times__int__code_I3_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,pos,M2)),aa(num,int,pos,N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ).

% times_int_code(3)
tff(fact_8071_of__int__code_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,pos,K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_code(3)
tff(fact_8072_Int_OPos__def,axiom,
    pos = numeral_numeral(int) ).

% Int.Pos_def
tff(fact_8073_plus__int__code_I3_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,pos,M2)),aa(num,int,pos,N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)) ).

% plus_int_code(3)
tff(fact_8074_less__eq__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,pos,K)),aa(num,int,pos,L)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),K),L)) ) ).

% less_eq_int_code(5)
tff(fact_8075_less__eq__int__code_I2_J,axiom,
    ! [L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(num,int,pos,L))) ).

% less_eq_int_code(2)
tff(fact_8076_less__eq__int__code_I4_J,axiom,
    ! [K: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,pos,K)),zero_zero(int))) ).

% less_eq_int_code(4)
tff(fact_8077_nat__code_I3_J,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(num,int,pos,K)) = aa(num,nat,nat_of_num,K) ).

% nat_code(3)
tff(fact_8078_less__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,pos,K)),aa(num,int,pos,L)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),K),L)) ) ).

% less_int_code(5)
tff(fact_8079_one__int__code,axiom,
    one_one(int) = aa(num,int,pos,one2) ).

% one_int_code
tff(fact_8080_less__int__code_I4_J,axiom,
    ! [K: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,pos,K)),zero_zero(int))) ).

% less_int_code(4)
tff(fact_8081_less__int__code_I2_J,axiom,
    ! [L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(num,int,pos,L))) ).

% less_int_code(2)
tff(fact_8082_minus__int__code_I3_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,pos,M2)),aa(num,int,pos,N)) = sub(M2,N) ).

% minus_int_code(3)
tff(fact_8083_minus__int__code_I4_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,pos,M2)),neg(N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)) ).

% minus_int_code(4)
tff(fact_8084_minus__int__code_I5_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),neg(M2)),aa(num,int,pos,N)) = neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)) ).

% minus_int_code(5)
tff(fact_8085_times__int__code_I4_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,pos,M2)),neg(N)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ).

% times_int_code(4)
tff(fact_8086_times__int__code_I5_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),neg(M2)),aa(num,int,pos,N)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ).

% times_int_code(5)
tff(fact_8087_times__int__code_I6_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),neg(M2)),neg(N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ).

% times_int_code(6)
tff(fact_8088_plus__int__code_I5_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),neg(M2)),aa(num,int,pos,N)) = sub(N,M2) ).

% plus_int_code(5)
tff(fact_8089_plus__int__code_I4_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,pos,M2)),neg(N)) = sub(M2,N) ).

% plus_int_code(4)
tff(fact_8090_is__arg__max__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F2: fun(B,A),P: fun(B,bool),X: B] :
          ( lattic501386751176901750rg_max(B,A,F2,P,X)
        <=> ( pp(aa(B,bool,P,X))
            & ~ ? [Y4: B] :
                  ( pp(aa(B,bool,P,Y4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X)),aa(B,A,F2,Y4))) ) ) ) ) ).

% is_arg_max_def
tff(fact_8091_prod__list_Ocomm__monoid__list__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups1828464146339083142d_list(A,times_times(A),one_one(A)) ) ).

% prod_list.comm_monoid_list_axioms
tff(fact_8092_sum__list_Ocomm__monoid__list__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups1828464146339083142d_list(A,plus_plus(A),zero_zero(A)) ) ).

% sum_list.comm_monoid_list_axioms
tff(fact_8093_VEBT_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B))),bool,aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A))),fun(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B))),bool),bNF_rel_fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B)),bNF_rel_fun(option(product_prod(nat,nat)),option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)))),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B)))),fequal(option(product_prod(nat,nat))),bNF_rel_fun(nat,nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))),fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))),fequal(nat),bNF_rel_fun(list(product_prod(vEBT_VEBT,A)),list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(A,A)),fun(vEBT_VEBT,fun(B,B)),list_all2(product_prod(vEBT_VEBT,A),product_prod(vEBT_VEBT,B),basic_rel_prod(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,fun(A,A),fun(B,B),fequal(vEBT_VEBT),bNF_rel_fun(A,B,A,B,S3,S3))))),bNF_rel_fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),fun(vEBT_VEBT,A),fun(vEBT_VEBT,B),bNF_rel_fun(bool,bool,fun(bool,A),fun(bool,B),fequal(bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),S3)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3))),vEBT_rec_VEBT(A)),vEBT_rec_VEBT(B))) ).

% VEBT.rec_transfer
tff(fact_8094_group_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z,Inverse)
     => group_axioms(A,F2,Z,Inverse) ) ).

% group.axioms(2)
tff(fact_8095_group__axioms_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,Z),A6) = A6
     => ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A6)),A6) = Z
       => group_axioms(A,F2,Z,Inverse) ) ) ).

% group_axioms.intro
tff(fact_8096_group__axioms__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group_axioms(A,F2,Z,Inverse)
    <=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,Z),A5) = A5
        & ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A5)),A5) = Z ) ) ).

% group_axioms_def
tff(fact_8097_VEBT_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B))),bool,aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A))),fun(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B))),bool),bNF_rel_fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B)),bNF_rel_fun(option(product_prod(nat,nat)),option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))),fequal(option(product_prod(nat,nat))),bNF_rel_fun(nat,nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)),fequal(nat),bNF_rel_fun(list(vEBT_VEBT),list(vEBT_VEBT),fun(vEBT_VEBT,A),fun(vEBT_VEBT,B),fequal(list(vEBT_VEBT)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3)))),bNF_rel_fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),fun(vEBT_VEBT,A),fun(vEBT_VEBT,B),bNF_rel_fun(bool,bool,fun(bool,A),fun(bool,B),fequal(bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),S3)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3))),vEBT_case_VEBT(A)),vEBT_case_VEBT(B))) ).

% VEBT.case_transfer
tff(fact_8098_group__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z,Inverse)
    <=> ( semigroup(A,F2)
        & group_axioms(A,F2,Z,Inverse) ) ) ).

% group_def
tff(fact_8099_monoid_Oaxioms_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( monoid(A,F2,Z)
     => semigroup(A,F2) ) ).

% monoid.axioms(1)
tff(fact_8100_max_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semigroup(A,ord_max(A)) ) ).

% max.semigroup_axioms
tff(fact_8101_VEBT_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F22: fun(bool,fun(bool,A)),VEBT: vEBT_VEBT] : aa(A,B,H,aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_case_VEBT(A),F1),F22),VEBT)) = aa(vEBT_VEBT,B,aa(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B)),vEBT_case_VEBT(B),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),aTP_Lamp_aid(fun(A,B),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))))),H),F1)),aa(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),aTP_Lamp_aie(fun(A,B),fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B))),H),F22)),VEBT) ).

% VEBT.case_distrib
tff(fact_8102_semigroup__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A))] :
      ( semigroup(A,F2)
    <=> ! [A5: A,B3: A,C4: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A5),B3)),C4) = aa(A,A,aa(A,fun(A,A),F2,A5),aa(A,A,aa(A,fun(A,A),F2,B3),C4)) ) ).

% semigroup_def
tff(fact_8103_semigroup_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A))] :
      ( ! [A6: A,B4: A,C3: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A6),B4)),C3) = aa(A,A,aa(A,fun(A,A),F2,A6),aa(A,A,aa(A,fun(A,A),F2,B4),C3))
     => semigroup(A,F2) ) ).

% semigroup.intro
tff(fact_8104_semigroup_Oassoc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
      ( semigroup(A,F2)
     => ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)) ) ) ).

% semigroup.assoc
tff(fact_8105_min_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semigroup(A,ord_min(A)) ) ).

% min.semigroup_axioms
tff(fact_8106_VEBT_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F22: fun(bool,fun(bool,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_case_VEBT(A),F1),F22),vEBT_Node(X11,X12,X13,X14)) = aa(vEBT_VEBT,A,aa(list(vEBT_VEBT),fun(vEBT_VEBT,A),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),F1,X11),X12),X13),X14) ).

% VEBT.simps(5)
tff(fact_8107_VEBT_Osimps_I6_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F22: fun(bool,fun(bool,A)),X21: bool,X22: bool] : aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_case_VEBT(A),F1),F22),vEBT_Leaf(X21,X22)) = aa(bool,A,aa(bool,fun(bool,A),F22,X21),X22) ).

% VEBT.simps(6)
tff(fact_8108_group_Oaxioms_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z,Inverse)
     => semigroup(A,F2) ) ).

% group.axioms(1)
tff(fact_8109_inf_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => semigroup(A,inf_inf(A)) ) ).

% inf.semigroup_axioms
tff(fact_8110_sup_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semigroup(A,sup_sup(A)) ) ).

% sup.semigroup_axioms
tff(fact_8111_add_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => semigroup(A,plus_plus(A)) ) ).

% add.semigroup_axioms
tff(fact_8112_mult_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => semigroup(A,times_times(A)) ) ).

% mult.semigroup_axioms
tff(fact_8113_group_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( semigroup(A,F2)
     => ( group_axioms(A,F2,Z,Inverse)
       => group(A,F2,Z,Inverse) ) ) ).

% group.intro
tff(fact_8114_monoid__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( monoid(A,F2,Z)
    <=> ( semigroup(A,F2)
        & monoid_axioms(A,F2,Z) ) ) ).

% monoid_def
tff(fact_8115_monoid_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( semigroup(A,F2)
     => ( monoid_axioms(A,F2,Z)
       => monoid(A,F2,Z) ) ) ).

% monoid.intro
tff(fact_8116_monoid__axioms_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,Z),A6) = A6
     => ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,A6),Z) = A6
       => monoid_axioms(A,F2,Z) ) ) ).

% monoid_axioms.intro
tff(fact_8117_monoid__axioms__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( monoid_axioms(A,F2,Z)
    <=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,Z),A5) = A5
        & ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,A5),Z) = A5 ) ) ).

% monoid_axioms_def
tff(fact_8118_monoid_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( monoid(A,F2,Z)
     => monoid_axioms(A,F2,Z) ) ).

% monoid.axioms(2)
tff(fact_8119_Quotient3__real,axiom,
    quotient3(fun(nat,rat),real,realrel,real2,rep_real) ).

% Quotient3_real
tff(fact_8120_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_nu(product_prod(int,int),product_prod(int,int)),Uu) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uu))) ).

% ATP.lambda_1
tff(fact_8121_ATP_Olambda__2,axiom,
    ! [Uu: fun(nat,rat)] : aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_abs(fun(nat,rat),fun(nat,rat)),Uu) = if(fun(nat,rat),vanishes(Uu),aTP_Lamp_abq(nat,rat),aTP_Lamp_abr(fun(nat,rat),fun(nat,rat),Uu)) ).

% ATP.lambda_2
tff(fact_8122_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_ba(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_3
tff(fact_8123_ATP_Olambda__4,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_un(A,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),Uu)),one_one(A))),Uu) ) ).

% ATP.lambda_4
tff(fact_8124_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_aag(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_5
tff(fact_8125_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_ok(A,bool),Uu))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uu),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Uu)) ) ) ) ).

% ATP.lambda_6
tff(fact_8126_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_cu(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_8127_ATP_Olambda__8,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_hp(real,bool),Uu))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Uu))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uu),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
        & ( aa(real,real,cos(real),Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_8
tff(fact_8128_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),A,aTP_Lamp_nt(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_8129_ATP_Olambda__10,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_vr(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_8130_ATP_Olambda__11,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_nv(product_prod(int,int),product_prod(int,int)),Uu) = aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(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_11
tff(fact_8131_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_dm(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,cos_coeff,Uu)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu)) ).

% ATP.lambda_12
tff(fact_8132_ATP_Olambda__13,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_vq(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_13
tff(fact_8133_ATP_Olambda__14,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_us(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,cos(real),Uu)),aa(real,real,sin(real),Uu)) ).

% ATP.lambda_14
tff(fact_8134_ATP_Olambda__15,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_xw(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Uu)),Uu) ).

% ATP.lambda_15
tff(fact_8135_ATP_Olambda__16,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_ahh(list(A),product_prod(nat,list(A))),Uu) = aa(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_8136_ATP_Olambda__17,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( pp(aa(nat,bool,aTP_Lamp_ma(nat,bool),Uu))
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_17
tff(fact_8137_ATP_Olambda__18,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_ve(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_18
tff(fact_8138_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_kz(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_19
tff(fact_8139_ATP_Olambda__20,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_hd(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_20
tff(fact_8140_ATP_Olambda__21,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahj(A,bool),Uu))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uu),one_one(A))) ) ) ).

% ATP.lambda_21
tff(fact_8141_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_oz(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uu),zero_zero(nat)) ).

% ATP.lambda_22
tff(fact_8142_ATP_Olambda__23,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( pp(aa(option(B),bool,aTP_Lamp_ahe(option(B),bool),Uu))
    <=> ( Uu = none(B) ) ) ).

% ATP.lambda_23
tff(fact_8143_ATP_Olambda__24,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_ahd(option(A),bool),Uu))
    <=> ( Uu = none(A) ) ) ).

% ATP.lambda_24
tff(fact_8144_ATP_Olambda__25,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_ns(product_prod(int,int),bool),Uu))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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_8145_ATP_Olambda__26,axiom,
    ! [Uu: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_ahv(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aTP_Lamp_ahu(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu)) ).

% ATP.lambda_26
tff(fact_8146_ATP_Olambda__27,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_vl(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_27
tff(fact_8147_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_vp(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_28
tff(fact_8148_ATP_Olambda__29,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ado(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_29
tff(fact_8149_ATP_Olambda__30,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( pp(aa(option(B),bool,aTP_Lamp_ahc(option(B),bool),Uu))
    <=> ( Uu != none(B) ) ) ).

% ATP.lambda_30
tff(fact_8150_ATP_Olambda__31,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_ahb(option(A),bool),Uu))
    <=> ( Uu != none(A) ) ) ).

% ATP.lambda_31
tff(fact_8151_ATP_Olambda__32,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_adp(list(B),bool),Uu))
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_32
tff(fact_8152_ATP_Olambda__33,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_adh(list(A),bool),Uu))
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_33
tff(fact_8153_ATP_Olambda__34,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_pa(real,real),Uu) = suminf(real,aTP_Lamp_bb(real,fun(nat,real),Uu)) ).

% ATP.lambda_34
tff(fact_8154_ATP_Olambda__35,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_adx(nat,set(nat)),Uu) = collect(nat,aTP_Lamp_jk(nat,fun(nat,bool),Uu)) ).

% ATP.lambda_35
tff(fact_8155_ATP_Olambda__36,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_vn(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_36
tff(fact_8156_ATP_Olambda__37,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_en(nat,A),Uu) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu)) ) ).

% ATP.lambda_37
tff(fact_8157_ATP_Olambda__38,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_vg(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_38
tff(fact_8158_ATP_Olambda__39,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_adn(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_39
tff(fact_8159_ATP_Olambda__40,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_adg(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_40
tff(fact_8160_ATP_Olambda__41,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_zt(A,B),Uu) = aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,Uu)) ) ).

% ATP.lambda_41
tff(fact_8161_ATP_Olambda__42,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_zu(A,B),Uu) = aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,Uu)) ) ).

% ATP.lambda_42
tff(fact_8162_ATP_Olambda__43,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_nj(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_43
tff(fact_8163_ATP_Olambda__44,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_nd(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).

% ATP.lambda_44
tff(fact_8164_ATP_Olambda__45,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kq(int,fun(int,product_prod(int,int))),Uu) = product_Pair(int,int,aa(int,int,uminus_uminus(int),Uu)) ).

% ATP.lambda_45
tff(fact_8165_ATP_Olambda__46,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_hs(nat,fun(nat,product_prod(nat,nat))),Uu) = product_Pair(nat,nat,aa(nat,nat,suc,Uu)) ).

% ATP.lambda_46
tff(fact_8166_ATP_Olambda__47,axiom,
    ! [Uu: nat] : aa(nat,extended_enat,aTP_Lamp_afv(nat,extended_enat),Uu) = extended_enat2(aa(nat,nat,suc,Uu)) ).

% ATP.lambda_47
tff(fact_8167_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_oi(A,bool),Uu))
        <=> ? [N5: int] :
              ( ( Uu = aa(int,A,ring_1_of_int(A),N5) )
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N5)) ) ) ) ).

% ATP.lambda_48
tff(fact_8168_ATP_Olambda__49,axiom,
    ! [Uu: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aTP_Lamp_abu(fun(nat,rat),bool),Uu))
    <=> ? [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
          & ? [K3: nat] :
            ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,Uu,N5))) ) ) ) ).

% ATP.lambda_49
tff(fact_8169_ATP_Olambda__50,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_of(real,bool),Uu))
    <=> ? [I2: int,N5: nat] :
          ( ( Uu = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),I2)),aa(nat,real,semiring_1_of_nat(real),N5)) )
          & ( N5 != zero_zero(nat) ) ) ) ).

% ATP.lambda_50
tff(fact_8170_ATP_Olambda__51,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_oe(real,bool),Uu))
    <=> ? [I2: int,J3: int] :
          ( ( Uu = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),I2)),aa(int,real,ring_1_of_int(real),J3)) )
          & ( J3 != zero_zero(int) ) ) ) ).

% ATP.lambda_51
tff(fact_8171_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yi(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),Y4) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4)) ) ) ) ).

% ATP.lambda_52
tff(fact_8172_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yh(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),Y4) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X4)) ) ) ) ).

% ATP.lambda_53
tff(fact_8173_ATP_Olambda__54,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_nb(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_54
tff(fact_8174_ATP_Olambda__55,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_nm(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_nk(nat,fun(num,option(num)),Uua)),aTP_Lamp_nl(nat,fun(num,option(num)),Uua)),Uu) ).

% ATP.lambda_55
tff(fact_8175_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hb(A,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_56
tff(fact_8176_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ek(nat,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),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_57
tff(fact_8177_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hc(A,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),zero_zero(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_58
tff(fact_8178_ATP_Olambda__59,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_cx(fun(nat,real),fun(nat,real),Uu),Uua) = if(real,aa(nat,bool,aa(nat,fun(nat,bool),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_59
tff(fact_8179_ATP_Olambda__60,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_ln(code_integer,fun(code_integer,int)),Uu),Uua) = if(int,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(code_integer,int,code_int_of_integer,Uu)),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))),aa(code_integer,int,code_int_of_integer,Uu))),one_one(int))) ).

% ATP.lambda_60
tff(fact_8180_ATP_Olambda__61,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ii(nat,fun(nat,A)),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uua),zero_zero(nat)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),Uu))),one_one(A))) ) ).

% ATP.lambda_61
tff(fact_8181_ATP_Olambda__62,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_nw(code_integer,fun(code_integer,num)),Uu),Uua) = if(num,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu)),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu))),one2)) ).

% ATP.lambda_62
tff(fact_8182_ATP_Olambda__63,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_lj(code_integer,fun(code_integer,nat)),Uu),Uua) = if(nat,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one(nat))) ).

% ATP.lambda_63
tff(fact_8183_ATP_Olambda__64,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kp(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uu),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_64
tff(fact_8184_ATP_Olambda__65,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ou(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,Uu),Uua)) ).

% ATP.lambda_65
tff(fact_8185_ATP_Olambda__66,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_afu(extended_enat,fun(nat,extended_enat),Uu),Uua) = extended_case_enat(extended_enat,aTP_Lamp_aft(nat,fun(nat,extended_enat),Uua),if(extended_enat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uua),zero_zero(nat)),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)),Uu) ).

% ATP.lambda_66
tff(fact_8186_ATP_Olambda__67,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_afs(extended_enat,fun(nat,extended_enat),Uu),Uua) = extended_case_enat(extended_enat,aTP_Lamp_afr(nat,fun(nat,extended_enat),Uua),zero_zero(extended_enat),Uu) ).

% ATP.lambda_67
tff(fact_8187_ATP_Olambda__68,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gi(nat,fun(nat,A),Uu),Uua) = if(A,aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),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_68
tff(fact_8188_ATP_Olambda__69,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aeq(extended_enat,fun(nat,bool),Uu),Uua))
    <=> pp(extended_case_enat(bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),fTrue,Uu)) ) ).

% ATP.lambda_69
tff(fact_8189_ATP_Olambda__70,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_zw(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_70
tff(fact_8190_ATP_Olambda__71,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_zv(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_71
tff(fact_8191_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_nn(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_nm(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_72
tff(fact_8192_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_nk(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_nd(num,option(num))),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_73
tff(fact_8193_ATP_Olambda__74,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_ng(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_nd(num,option(num))),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_74
tff(fact_8194_ATP_Olambda__75,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_kk(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua))) ) ).

% ATP.lambda_75
tff(fact_8195_ATP_Olambda__76,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_acx(list(list(A)),fun(nat,list(A)),Uu),Uua) = map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_acw(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_76
tff(fact_8196_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hx(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_hw(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_77
tff(fact_8197_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hv(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_78
tff(fact_8198_ATP_Olambda__79,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ap(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_79
tff(fact_8199_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ca(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_80
tff(fact_8200_ATP_Olambda__81,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_dd(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_81
tff(fact_8201_ATP_Olambda__82,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fz(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_82
tff(fact_8202_ATP_Olambda__83,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ga(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_83
tff(fact_8203_ATP_Olambda__84,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_hr(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
        & ( aa(real,real,sin(real),Uua) = Uu ) ) ) ).

% ATP.lambda_84
tff(fact_8204_ATP_Olambda__85,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_hh(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),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_85
tff(fact_8205_ATP_Olambda__86,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_hf(complex,fun(real,bool),Uu),Uua))
    <=> ( ( aa(complex,complex,sgn_sgn(complex),Uu) = cis(Uua) )
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),pi)) ) ) ).

% ATP.lambda_86
tff(fact_8206_ATP_Olambda__87,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_kl(real,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Uu),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_87
tff(fact_8207_ATP_Olambda__88,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_km(rat,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu))
        & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_88
tff(fact_8208_ATP_Olambda__89,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bb(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_8209_ATP_Olambda__90,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_pb(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_8210_ATP_Olambda__91,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ux(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_8211_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_8212_ATP_Olambda__93,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_oa(rat,fun(product_prod(int,int),bool),Uu),Uua))
    <=> ( ( Uu = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uua)) )
        & pp(aa(int,bool,aa(int,fun(int,bool),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_8213_ATP_Olambda__94,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fb(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_8214_ATP_Olambda__95,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_de(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_8215_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fl(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_8216_ATP_Olambda__97,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lm(nat,fun(nat,bool)),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_97
tff(fact_8217_ATP_Olambda__98,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_aah(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_98
tff(fact_8218_ATP_Olambda__99,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_ni(set(option(A)),fun(option(A),bool),Uu),Uua))
    <=> ( pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),Uua),Uu))
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_99
tff(fact_8219_ATP_Olambda__100,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ee(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_100
tff(fact_8220_ATP_Olambda__101,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_yc(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu)),aa(real,real,exp(real),Uua)) ).

% ATP.lambda_101
tff(fact_8221_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aff(nat,fun(nat,set(nat)),Uu),Uua) = set_or3652927894154168847AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua),Uu) ).

% ATP.lambda_102
tff(fact_8222_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zx(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_103
tff(fact_8223_ATP_Olambda__104,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ex(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_104
tff(fact_8224_ATP_Olambda__105,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ew(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_105
tff(fact_8225_ATP_Olambda__106,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa(complex,bool,aTP_Lamp_bh(nat,fun(complex,bool),Uu),Uua))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_106
tff(fact_8226_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_jn(nat,fun(A,bool),Uu),Uua))
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_107
tff(fact_8227_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_jl(A,fun(A,bool),Uu),Uua))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu)) ) ) ) ).

% ATP.lambda_108
tff(fact_8228_ATP_Olambda__109,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_vx(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_109
tff(fact_8229_ATP_Olambda__110,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_yk(real,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Uu),Uua))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Uua)) ).

% ATP.lambda_110
tff(fact_8230_ATP_Olambda__111,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_ye(real,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),Uua))),Uua) ).

% ATP.lambda_111
tff(fact_8231_ATP_Olambda__112,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_an(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_112
tff(fact_8232_ATP_Olambda__113,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_hm(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),pi))
        & ( aa(real,real,cos(real),Uua) = Uu ) ) ) ).

% ATP.lambda_113
tff(fact_8233_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: set(A),Uua: list(A)] :
          ( pp(aa(list(A),bool,aTP_Lamp_agd(set(A),fun(list(A),bool),Uu),Uua))
        <=> ( sorted_wrt(A,ord_less(A),Uua)
            & ( aa(list(A),set(A),set2(A),Uua) = Uu ) ) ) ) ).

% ATP.lambda_114
tff(fact_8234_ATP_Olambda__115,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_agb(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ) ).

% ATP.lambda_115
tff(fact_8235_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_agc(nat,fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu))) ) ) ).

% ATP.lambda_116
tff(fact_8236_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ay(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_117
tff(fact_8237_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cw(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_118
tff(fact_8238_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vi(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_119
tff(fact_8239_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cp(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_8240_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ic(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_121
tff(fact_8241_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ia(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_122
tff(fact_8242_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bt(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_8243_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_as(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_8244_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_iv(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_125
tff(fact_8245_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bu(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_126
tff(fact_8246_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ct(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_127
tff(fact_8247_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cs(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),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_128
tff(fact_8248_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vj(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_129
tff(fact_8249_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ds(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_130
tff(fact_8250_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_et(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_131
tff(fact_8251_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_eq(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_132
tff(fact_8252_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,A,aTP_Lamp_aee(fun(A,real),fun(A,A),Uu),Uua) = real_V8093663219630862766scaleR(A,aa(A,real,Uu,Uua),Uua) ) ).

% ATP.lambda_133
tff(fact_8253_ATP_Olambda__134,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_ji(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = zero_zero(A) ) ) ) ).

% ATP.lambda_134
tff(fact_8254_ATP_Olambda__135,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: fun(A,B),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aic(fun(A,B),fun(A,bool),Uu),Uua))
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_135
tff(fact_8255_ATP_Olambda__136,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_jj(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).

% ATP.lambda_136
tff(fact_8256_ATP_Olambda__137,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_wh(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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_137
tff(fact_8257_ATP_Olambda__138,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_wg(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ux(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_138
tff(fact_8258_ATP_Olambda__139,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_acz(list(A),fun(list(A),list(list(A))),Uu),Uua) = map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_acy(list(A),fun(A,list(A))),Uua),Uu) ).

% ATP.lambda_139
tff(fact_8259_ATP_Olambda__140,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_mb(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_140
tff(fact_8260_ATP_Olambda__141,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_xh(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_141
tff(fact_8261_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gp(A,fun(nat,A),Uu),Uua) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,A,aa(A,fun(nat,A),power_power(A),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_142
tff(fact_8262_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gz(A,fun(nat,A),Uu),Uua) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_143
tff(fact_8263_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_agf(set(A),fun(set(A),bool),Uu),Uua))
        <=> ( ~ real_V358717886546972837endent(A,Uua)
            & ( real_Vector_span(A,Uua) = real_Vector_span(A,Uu) ) ) ) ) ).

% ATP.lambda_144
tff(fact_8264_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gq(A,fun(nat,A),Uu),Uua) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_145
tff(fact_8265_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_by(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_146
tff(fact_8266_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_df(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_147
tff(fact_8267_ATP_Olambda__148,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_mz(num,fun(num,int),Uu),Uua) = 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_148
tff(fact_8268_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gx(A,fun(nat,A),Uu),Uua) = real_V8093663219630862766scaleR(A,aa(nat,real,cos_coeff,Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua)) ) ).

% ATP.lambda_149
tff(fact_8269_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_oy(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_150
tff(fact_8270_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gr(A,fun(nat,A),Uu),Uua) = real_V8093663219630862766scaleR(A,aa(nat,real,sin_coeff,Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_151
tff(fact_8271_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gs(A,fun(nat,A),Uu),Uua) = real_V8093663219630862766scaleR(A,aa(nat,real,cos_coeff,Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_152
tff(fact_8272_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_wc(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_153
tff(fact_8273_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_wb(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_154
tff(fact_8274_ATP_Olambda__155,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_dl(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,sin_coeff,Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_155
tff(fact_8275_ATP_Olambda__156,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_dk(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,cos_coeff,Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_156
tff(fact_8276_ATP_Olambda__157,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_abg(A,fun(set(A),bool),Uu),Uua))
        <=> ( topolo1002775350975398744n_open(A,Uua)
            & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uu),Uua)) ) ) ) ).

% ATP.lambda_157
tff(fact_8277_ATP_Olambda__158,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(product_prod(A,B))] :
      ( pp(aa(option(product_prod(A,B)),bool,aTP_Lamp_agx(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),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)),collect(product_prod(A,B),product_case_prod(A,B,bool,Uu)))) ) ).

% ATP.lambda_158
tff(fact_8278_ATP_Olambda__159,axiom,
    ! [A: $tType,Uu: list(option(A)),Uua: A] : aa(A,option(list(A)),aTP_Lamp_ahm(list(option(A)),fun(A,option(list(A))),Uu),Uua) = aa(option(list(A)),option(list(A)),aa(fun(list(A),list(A)),fun(option(list(A)),option(list(A))),map_option(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Uua)),those(A,Uu)) ).

% ATP.lambda_159
tff(fact_8279_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lr(nat,fun(nat,A)),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_160
tff(fact_8280_ATP_Olambda__161,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_ot(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_161
tff(fact_8281_ATP_Olambda__162,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_og(list(A),fun(list(A),bool)),Uu),Uua))
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_162
tff(fact_8282_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_cz(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_163
tff(fact_8283_ATP_Olambda__164,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_adl(set(nat),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)),Uu)) ) ).

% ATP.lambda_164
tff(fact_8284_ATP_Olambda__165,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_acn(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,Uua)),Uu)) ) ).

% ATP.lambda_165
tff(fact_8285_ATP_Olambda__166,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_hk(real,fun(real,bool),Uu),Uua))
    <=> ( aa(real,real,exp(real),Uua) = Uu ) ) ).

% ATP.lambda_166
tff(fact_8286_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_hg(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(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_167
tff(fact_8287_ATP_Olambda__168,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ao(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_168
tff(fact_8288_ATP_Olambda__169,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_169
tff(fact_8289_ATP_Olambda__170,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_vz(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))))) ).

% ATP.lambda_170
tff(fact_8290_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vu(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_171
tff(fact_8291_ATP_Olambda__172,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_afq(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_172
tff(fact_8292_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ie(A,fun(nat,A),Uu),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_173
tff(fact_8293_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_vv(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_174
tff(fact_8294_ATP_Olambda__175,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_aer(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int))))) ).

% ATP.lambda_175
tff(fact_8295_ATP_Olambda__176,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_vw(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua))))) ).

% ATP.lambda_176
tff(fact_8296_ATP_Olambda__177,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_vo(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_177
tff(fact_8297_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( 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),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_178
tff(fact_8298_ATP_Olambda__179,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_adb(nat,fun(list(A),bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua))) ) ).

% ATP.lambda_179
tff(fact_8299_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_az(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_180
tff(fact_8300_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ih(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_181
tff(fact_8301_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aw(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_182
tff(fact_8302_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_aev(nat,fun(nat,list(nat))),Uu),Uua) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Uu),aa(nat,list(nat),nat_list_decode,Uua)) ).

% ATP.lambda_183
tff(fact_8303_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aep(nat,fun(nat,bool)),Uu),Uua))
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_184
tff(fact_8304_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat,Uua: A] : aa(A,A,aTP_Lamp_uj(nat,fun(A,A),Uu),Uua) = comm_s3205402744901411588hammer(A,Uua,Uu) ) ).

% ATP.lambda_185
tff(fact_8305_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lb(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uu)) ) ).

% ATP.lambda_186
tff(fact_8306_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_acb(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_187
tff(fact_8307_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aey(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_188
tff(fact_8308_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aei(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_189
tff(fact_8309_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ahq(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_190
tff(fact_8310_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aew(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_191
tff(fact_8311_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_or(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_192
tff(fact_8312_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wn(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_193
tff(fact_8313_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mi(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_194
tff(fact_8314_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aay(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_195
tff(fact_8315_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).

% ATP.lambda_196
tff(fact_8316_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_acc(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_197
tff(fact_8317_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_yl(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_198
tff(fact_8318_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aez(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_199
tff(fact_8319_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aej(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_200
tff(fact_8320_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aeo(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_201
tff(fact_8321_ATP_Olambda__202,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aex(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_202
tff(fact_8322_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_dn(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_203
tff(fact_8323_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_vd(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_204
tff(fact_8324_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mr(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_205
tff(fact_8325_ATP_Olambda__206,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_afo(real,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),minus_minus(real),Uua),Uu) ).

% ATP.lambda_206
tff(fact_8326_ATP_Olambda__207,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_207
tff(fact_8327_ATP_Olambda__208,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_afn(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_208
tff(fact_8328_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mh(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_209
tff(fact_8329_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mj(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_210
tff(fact_8330_ATP_Olambda__211,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_pt(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_211
tff(fact_8331_ATP_Olambda__212,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_abx(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_212
tff(fact_8332_ATP_Olambda__213,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aaa(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).

% ATP.lambda_213
tff(fact_8333_ATP_Olambda__214,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ada(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_214
tff(fact_8334_ATP_Olambda__215,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_nc(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_215
tff(fact_8335_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_afp(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_216
tff(fact_8336_ATP_Olambda__217,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_pw(real,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),powr(real),Uua),Uu) ).

% ATP.lambda_217
tff(fact_8337_ATP_Olambda__218,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_jk(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uua),Uu)) ) ).

% ATP.lambda_218
tff(fact_8338_ATP_Olambda__219,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_ka(int,fun(int,bool),Uu),Uua))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uua),Uu)) ) ).

% ATP.lambda_219
tff(fact_8339_ATP_Olambda__220,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ac(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),Uu)) ) ) ).

% ATP.lambda_220
tff(fact_8340_ATP_Olambda__221,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_lq(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uua),Uu) ).

% ATP.lambda_221
tff(fact_8341_ATP_Olambda__222,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_eo(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_222
tff(fact_8342_ATP_Olambda__223,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_acy(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_223
tff(fact_8343_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : aa(A,A,aTP_Lamp_abh(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ).

% ATP.lambda_224
tff(fact_8344_ATP_Olambda__225,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_vm(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_225
tff(fact_8345_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aec(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_226
tff(fact_8346_ATP_Olambda__227,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_a(set(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_227
tff(fact_8347_ATP_Olambda__228,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_aga(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_228
tff(fact_8348_ATP_Olambda__229,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_adc(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_229
tff(fact_8349_ATP_Olambda__230,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_hi(A,fun(A,bool),Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_230
tff(fact_8350_ATP_Olambda__231,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_xb(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_231
tff(fact_8351_ATP_Olambda__232,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_wy(fun(B,real),fun(B,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(B,real,Uu,Uua))) ) ).

% ATP.lambda_232
tff(fact_8352_ATP_Olambda__233,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_wv(fun(A,real),fun(A,bool),Uu),Uua))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(A,real,Uu,Uua))) ) ) ).

% ATP.lambda_233
tff(fact_8353_ATP_Olambda__234,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_yb(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_234
tff(fact_8354_ATP_Olambda__235,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_agn(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_235
tff(fact_8355_ATP_Olambda__236,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ed(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))) ).

% ATP.lambda_236
tff(fact_8356_ATP_Olambda__237,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ec(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)) ).

% ATP.lambda_237
tff(fact_8357_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_yg(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_238
tff(fact_8358_ATP_Olambda__239,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_adi(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)))) ) ).

% ATP.lambda_239
tff(fact_8359_ATP_Olambda__240,axiom,
    ! [A: $tType,Uu: fun(int,A),Uua: nat] : aa(nat,A,aTP_Lamp_xs(fun(int,A),fun(nat,A),Uu),Uua) = aa(int,A,Uu,aa(nat,int,semiring_1_of_nat(int),Uua)) ).

% ATP.lambda_240
tff(fact_8360_ATP_Olambda__241,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_pe(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_241
tff(fact_8361_ATP_Olambda__242,axiom,
    ! [Uu: fun(real,bool),Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_yo(fun(real,bool),fun(real,bool),Uu),Uua))
    <=> pp(aa(real,bool,Uu,aa(real,real,uminus_uminus(real),Uua))) ) ).

% ATP.lambda_242
tff(fact_8362_ATP_Olambda__243,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_yn(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_243
tff(fact_8363_ATP_Olambda__244,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_adj(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_244
tff(fact_8364_ATP_Olambda__245,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_wl(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_245
tff(fact_8365_ATP_Olambda__246,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_246
tff(fact_8366_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_247
tff(fact_8367_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_248
tff(fact_8368_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_at(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_249
tff(fact_8369_ATP_Olambda__250,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ck(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_250
tff(fact_8370_ATP_Olambda__251,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_251
tff(fact_8371_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_ads(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_252
tff(fact_8372_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_adt(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_253
tff(fact_8373_ATP_Olambda__254,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_ady(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_254
tff(fact_8374_ATP_Olambda__255,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_adz(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_255
tff(fact_8375_ATP_Olambda__256,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_nl(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_256
tff(fact_8376_ATP_Olambda__257,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_nh(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_257
tff(fact_8377_ATP_Olambda__258,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_afz(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_afy(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_258
tff(fact_8378_ATP_Olambda__259,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ace(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = product_case_prod(nat,nat,bool,aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_acd(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_259
tff(fact_8379_ATP_Olambda__260,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_lz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_ly(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_260
tff(fact_8380_ATP_Olambda__261,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_lx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lw(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_261
tff(fact_8381_ATP_Olambda__262,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_lv(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = product_case_prod(nat,nat,bool,aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_lu(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_262
tff(fact_8382_ATP_Olambda__263,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_lt(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = product_case_prod(nat,nat,bool,aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_ls(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_263
tff(fact_8383_ATP_Olambda__264,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_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lo(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_264
tff(fact_8384_ATP_Olambda__265,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_qf(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_qe(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_265
tff(fact_8385_ATP_Olambda__266,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_pu(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_ey(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_266
tff(fact_8386_ATP_Olambda__267,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_aaw(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_aat(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),top_top(set(B)))) ) ).

% ATP.lambda_267
tff(fact_8387_ATP_Olambda__268,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_aas(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_aar(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),top_top(set(C)))) ) ).

% ATP.lambda_268
tff(fact_8388_ATP_Olambda__269,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_aau(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_aat(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),top_top(set(B)))) ) ).

% ATP.lambda_269
tff(fact_8389_ATP_Olambda__270,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_aav(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_aar(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),top_top(set(C)))) ) ).

% ATP.lambda_270
tff(fact_8390_ATP_Olambda__271,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_ej(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_271
tff(fact_8391_ATP_Olambda__272,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_rh(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_272
tff(fact_8392_ATP_Olambda__273,axiom,
    ! [Uu: fun(real,real),Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_xo(fun(real,real),fun(real,bool),Uu),Uua))
    <=> ( aa(real,real,Uu,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_273
tff(fact_8393_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aef(fun(A,real),fun(A,bool),Uu),Uua))
        <=> ( aa(A,real,Uu,Uua) != zero_zero(real) ) ) ) ).

% ATP.lambda_274
tff(fact_8394_ATP_Olambda__275,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aci(fun(A,option(B)),fun(A,bool),Uu),Uua))
    <=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).

% ATP.lambda_275
tff(fact_8395_ATP_Olambda__276,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_gv(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_276
tff(fact_8396_ATP_Olambda__277,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ak(nat,fun(nat,bool),Uu),Uua))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),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_277
tff(fact_8397_ATP_Olambda__278,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gw(A,fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(nat,real,sin_coeff,Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua))) ) ).

% ATP.lambda_278
tff(fact_8398_ATP_Olambda__279,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_gt(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,real_V8093663219630862766scaleR(A,aa(nat,real,sin_coeff,Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_279
tff(fact_8399_ATP_Olambda__280,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_gu(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,real_V8093663219630862766scaleR(A,aa(nat,real,cos_coeff,Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_280
tff(fact_8400_ATP_Olambda__281,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_ahu(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = product_Pair(code_natural,product_prod(code_natural,code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),Uu),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_281
tff(fact_8401_ATP_Olambda__282,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_aae(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),Uu,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_282
tff(fact_8402_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_am(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_283
tff(fact_8403_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_al(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_284
tff(fact_8404_ATP_Olambda__285,axiom,
    ! [A: $tType,Uu: list(A),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_ahx(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = product_Pair(A,product_prod(code_natural,code_natural),aa(nat,A,nth(A,Uu),aa(code_natural,nat,code_nat_of_natural,Uua))) ).

% ATP.lambda_285
tff(fact_8405_ATP_Olambda__286,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_aap(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,Uu,Uua)) ) ).

% ATP.lambda_286
tff(fact_8406_ATP_Olambda__287,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_aao(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8407_ATP_Olambda__288,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_afb(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_288
tff(fact_8408_ATP_Olambda__289,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_vt(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_289
tff(fact_8409_ATP_Olambda__290,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_pq(A,fun(A,A),Uu),Uua) = aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_290
tff(fact_8410_ATP_Olambda__291,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_ahw(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = product_Pair(code_natural,product_prod(code_natural,code_natural),modulo_modulo(code_natural,Uua,Uu)) ).

% ATP.lambda_291
tff(fact_8411_ATP_Olambda__292,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_aft(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uua)) ).

% ATP.lambda_292
tff(fact_8412_ATP_Olambda__293,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_afr(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua)) ).

% ATP.lambda_293
tff(fact_8413_ATP_Olambda__294,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_abe(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uu,Uua)) ) ).

% ATP.lambda_294
tff(fact_8414_ATP_Olambda__295,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_abd(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uua,Uu)) ) ).

% ATP.lambda_295
tff(fact_8415_ATP_Olambda__296,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_nx(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_296
tff(fact_8416_ATP_Olambda__297,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ny(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_297
tff(fact_8417_ATP_Olambda__298,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kn(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_298
tff(fact_8418_ATP_Olambda__299,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ko(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_299
tff(fact_8419_ATP_Olambda__300,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aa(set(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_300
tff(fact_8420_ATP_Olambda__301,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ade(A,fun(A,bool),Uu),Uua))
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_301
tff(fact_8421_ATP_Olambda__302,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_ff(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_302
tff(fact_8422_ATP_Olambda__303,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_sq(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_303
tff(fact_8423_ATP_Olambda__304,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: B] : aa(B,A,aTP_Lamp_mf(fun(B,bool),fun(B,A),Uu),Uua) = aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_304
tff(fact_8424_ATP_Olambda__305,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_afc(fun(B,A),fun(B,set(A)),Uu),Uua) = set_ord_atMost(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_305
tff(fact_8425_ATP_Olambda__306,axiom,
    ! [Uu: fun(nat,rat),Uua: nat] : aa(nat,rat,aTP_Lamp_abr(fun(nat,rat),fun(nat,rat),Uu),Uua) = aa(rat,rat,inverse_inverse(rat),aa(nat,rat,Uu,Uua)) ).

% ATP.lambda_306
tff(fact_8426_ATP_Olambda__307,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_rf(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_307
tff(fact_8427_ATP_Olambda__308,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_sp(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_308
tff(fact_8428_ATP_Olambda__309,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yx(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_309
tff(fact_8429_ATP_Olambda__310,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_za(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_310
tff(fact_8430_ATP_Olambda__311,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pf(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_311
tff(fact_8431_ATP_Olambda__312,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_uh(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_312
tff(fact_8432_ATP_Olambda__313,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yd(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_313
tff(fact_8433_ATP_Olambda__314,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] : aa(B,int,aTP_Lamp_be(fun(B,nat),fun(B,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(B,nat,Uu,Uua)) ).

% ATP.lambda_314
tff(fact_8434_ATP_Olambda__315,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_aq(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_315
tff(fact_8435_ATP_Olambda__316,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_cc(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_316
tff(fact_8436_ATP_Olambda__317,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rm(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_317
tff(fact_8437_ATP_Olambda__318,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_318
tff(fact_8438_ATP_Olambda__319,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ul(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_319
tff(fact_8439_ATP_Olambda__320,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_js(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_320
tff(fact_8440_ATP_Olambda__321,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_el(fun(nat,real),fun(nat,A),Uu),Uua) = real_Vector_of_real(A,aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_321
tff(fact_8441_ATP_Olambda__322,axiom,
    ! [Uu: fun(nat,rat),Uua: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ov(fun(nat,rat),fun(nat,rat)),Uu),Uua) = aa(rat,rat,uminus_uminus(rat),aa(nat,rat,Uu,Uua)) ).

% ATP.lambda_322
tff(fact_8442_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_323
tff(fact_8443_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_324
tff(fact_8444_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_he(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_325
tff(fact_8445_ATP_Olambda__326,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_em(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_326
tff(fact_8446_ATP_Olambda__327,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(C,B),Uua: C] : aa(C,B,aTP_Lamp_zj(fun(C,B),fun(C,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(C,B,Uu,Uua)) ) ).

% ATP.lambda_327
tff(fact_8447_ATP_Olambda__328,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_aai(fun(B,set(A)),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_328
tff(fact_8448_ATP_Olambda__329,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_aam(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_329
tff(fact_8449_ATP_Olambda__330,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_so(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_330
tff(fact_8450_ATP_Olambda__331,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ch(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_331
tff(fact_8451_ATP_Olambda__332,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_qw(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_332
tff(fact_8452_ATP_Olambda__333,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pg(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_333
tff(fact_8453_ATP_Olambda__334,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_agg(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_334
tff(fact_8454_ATP_Olambda__335,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ug(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_335
tff(fact_8455_ATP_Olambda__336,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_st(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_336
tff(fact_8456_ATP_Olambda__337,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ym(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,uminus_uminus(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_337
tff(fact_8457_ATP_Olambda__338,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ahs(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_338
tff(fact_8458_ATP_Olambda__339,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_sn(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_339
tff(fact_8459_ATP_Olambda__340,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_av(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_340
tff(fact_8460_ATP_Olambda__341,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_ar(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_341
tff(fact_8461_ATP_Olambda__342,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_cd(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_342
tff(fact_8462_ATP_Olambda__343,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_cq(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_343
tff(fact_8463_ATP_Olambda__344,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_zs(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,artanh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_344
tff(fact_8464_ATP_Olambda__345,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_345
tff(fact_8465_ATP_Olambda__346,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ua(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_346
tff(fact_8466_ATP_Olambda__347,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_zd(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arsinh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_347
tff(fact_8467_ATP_Olambda__348,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_tb(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,arsinh(real),aa(B,real,Uu,Uua)) ).

% ATP.lambda_348
tff(fact_8468_ATP_Olambda__349,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tg(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_349
tff(fact_8469_ATP_Olambda__350,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ry(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_350
tff(fact_8470_ATP_Olambda__351,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ze(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_351
tff(fact_8471_ATP_Olambda__352,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_352
tff(fact_8472_ATP_Olambda__353,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ta(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ).

% ATP.lambda_353
tff(fact_8473_ATP_Olambda__354,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_354
tff(fact_8474_ATP_Olambda__355,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_355
tff(fact_8475_ATP_Olambda__356,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_zo(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_356
tff(fact_8476_ATP_Olambda__357,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_tt(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,arcosh(real),aa(B,real,Uu,Uua)) ).

% ATP.lambda_357
tff(fact_8477_ATP_Olambda__358,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ww(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_358
tff(fact_8478_ATP_Olambda__359,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_359
tff(fact_8479_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_360
tff(fact_8480_ATP_Olambda__361,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ss(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,sgn_sgn(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_361
tff(fact_8481_ATP_Olambda__362,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zk(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_362
tff(fact_8482_ATP_Olambda__363,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ui(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_363
tff(fact_8483_ATP_Olambda__364,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ce(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_364
tff(fact_8484_ATP_Olambda__365,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [Uu: fun(A11,A11),Uua: A11] : aa(A11,A11,aTP_Lamp_qc(fun(A11,A11),fun(A11,A11),Uu),Uua) = aa(A11,A11,tanh(A11),aa(A11,A11,Uu,Uua)) ) ).

% ATP.lambda_365
tff(fact_8485_ATP_Olambda__366,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_zn(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_366
tff(fact_8486_ATP_Olambda__367,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_up(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_367
tff(fact_8487_ATP_Olambda__368,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_th(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_368
tff(fact_8488_ATP_Olambda__369,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [Uu: fun(A11,A11),Uua: A11] : aa(A11,A11,aTP_Lamp_pn(fun(A11,A11),fun(A11,A11),Uu),Uua) = aa(A11,A11,sinh(A11),aa(A11,A11,Uu,Uua)) ) ).

% ATP.lambda_369
tff(fact_8489_ATP_Olambda__370,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_zh(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,sinh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_370
tff(fact_8490_ATP_Olambda__371,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_sv(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,sinh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_371
tff(fact_8491_ATP_Olambda__372,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_te(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,sinh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_372
tff(fact_8492_ATP_Olambda__373,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rc(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,sinh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_373
tff(fact_8493_ATP_Olambda__374,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [Uu: fun(A11,A11),Uua: A11] : aa(A11,A11,aTP_Lamp_pm(fun(A11,A11),fun(A11,A11),Uu),Uua) = aa(A11,A11,cosh(A11),aa(A11,A11,Uu,Uua)) ) ).

% ATP.lambda_374
tff(fact_8494_ATP_Olambda__375,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_zf(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,cosh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_375
tff(fact_8495_ATP_Olambda__376,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_sw(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,cosh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_376
tff(fact_8496_ATP_Olambda__377,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_tc(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,cosh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_377
tff(fact_8497_ATP_Olambda__378,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rb(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cosh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_378
tff(fact_8498_ATP_Olambda__379,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_379
tff(fact_8499_ATP_Olambda__380,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_sl(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_380
tff(fact_8500_ATP_Olambda__381,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sin(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_381
tff(fact_8501_ATP_Olambda__382,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_sz(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sin(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_382
tff(fact_8502_ATP_Olambda__383,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pd(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,sin(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_383
tff(fact_8503_ATP_Olambda__384,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_zg(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,exp(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_384
tff(fact_8504_ATP_Olambda__385,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_sx(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,exp(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_385
tff(fact_8505_ATP_Olambda__386,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_td(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,exp(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_386
tff(fact_8506_ATP_Olambda__387,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qx(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,exp(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_387
tff(fact_8507_ATP_Olambda__388,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pc(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,exp(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_388
tff(fact_8508_ATP_Olambda__389,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_jm(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_389
tff(fact_8509_ATP_Olambda__390,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_su(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_390
tff(fact_8510_ATP_Olambda__391,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_wf(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,cos(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_391
tff(fact_8511_ATP_Olambda__392,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ri(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,cos(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_392
tff(fact_8512_ATP_Olambda__393,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_sy(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,cos(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_393
tff(fact_8513_ATP_Olambda__394,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pp(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cos(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_394
tff(fact_8514_ATP_Olambda__395,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_nq(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_395
tff(fact_8515_ATP_Olambda__396,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_qb(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_396
tff(fact_8516_ATP_Olambda__397,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_uq(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_397
tff(fact_8517_ATP_Olambda__398,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_398
tff(fact_8518_ATP_Olambda__399,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_lg(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_399
tff(fact_8519_ATP_Olambda__400,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ab(fun(A,bool),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(A,bool,Uu,Uua)) ) ).

% ATP.lambda_400
tff(fact_8520_ATP_Olambda__401,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_agl(nat,fun(nat,set(nat)),Uu),Uua) = collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_cv(nat,fun(nat,bool)),Uu)) ).

% ATP.lambda_401
tff(fact_8521_ATP_Olambda__402,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_oc(list(A),fun(A,bool),Uu),Uua))
    <=> ? [I2: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I2) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).

% ATP.lambda_402
tff(fact_8522_ATP_Olambda__403,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
      ( pp(aa(set(B),bool,aTP_Lamp_aaq(set(set(B)),fun(set(B),bool),Uu),Uua))
    <=> ? [F5: fun(set(B),B)] :
          ( ( Uua = image(set(B),B,F5,Uu) )
          & ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X4),Uu))
             => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(set(B),B,F5,X4)),X4)) ) ) ) ).

% ATP.lambda_403
tff(fact_8523_ATP_Olambda__404,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_aal(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image(set(A),A,F5,Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F5,X4)),X4)) ) ) ) ) ).

% ATP.lambda_404
tff(fact_8524_ATP_Olambda__405,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_afx(set(A),fun(set(A),bool),Uu),Uua))
    <=> ? [B8: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B8) )
          & pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Uu),pow2(A,B8))) ) ) ).

% ATP.lambda_405
tff(fact_8525_ATP_Olambda__406,axiom,
    ! [Uu: set(real),Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_aed(set(real),fun(real,bool),Uu),Uua))
    <=> ! [X4: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),Uu))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),Uua)) ) ) ).

% ATP.lambda_406
tff(fact_8526_ATP_Olambda__407,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_afe(nat,fun(nat,set(nat)),Uu),Uua) = set_ord_atMost(nat,Uu) ).

% ATP.lambda_407
tff(fact_8527_ATP_Olambda__408,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,rat,aTP_Lamp_ach(nat,fun(nat,rat),Uu),Uua) = aa(nat,rat,semiring_1_of_nat(rat),Uu) ).

% ATP.lambda_408
tff(fact_8528_ATP_Olambda__409,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_afh(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).

% ATP.lambda_409
tff(fact_8529_ATP_Olambda__410,axiom,
    ! [Uu: int,Uua: nat] : aa(nat,rat,aTP_Lamp_acg(int,fun(nat,rat),Uu),Uua) = aa(int,rat,ring_1_of_int(rat),Uu) ).

% ATP.lambda_410
tff(fact_8530_ATP_Olambda__411,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_aek(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_411
tff(fact_8531_ATP_Olambda__412,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(A),Uub: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aTP_Lamp_aha(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),Uu),Uua),Uub))
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),aa(bool,fun(fun(B,bool),fun(option(B),bool)),case_option(bool,B),fTrue),aTP_Lamp_agy(B,bool)),Uub)),aa(option(B),fun(A,bool),aTP_Lamp_agz(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_412
tff(fact_8532_ATP_Olambda__413,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_cy(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = if(real,aa(nat,bool,aa(nat,fun(nat,bool),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_413
tff(fact_8533_ATP_Olambda__414,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_eb(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = if(real,aa(nat,bool,aa(nat,fun(nat,bool),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_414
tff(fact_8534_ATP_Olambda__415,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_io(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),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),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_415
tff(fact_8535_ATP_Olambda__416,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_hn(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(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),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_416
tff(fact_8536_ATP_Olambda__417,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ho(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(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),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_417
tff(fact_8537_ATP_Olambda__418,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_hq(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(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),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_418
tff(fact_8538_ATP_Olambda__419,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(nat),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_iu(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_419
tff(fact_8539_ATP_Olambda__420,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_mk(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_420
tff(fact_8540_ATP_Olambda__421,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_ml(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_421
tff(fact_8541_ATP_Olambda__422,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_iq(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_422
tff(fact_8542_ATP_Olambda__423,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ir(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_423
tff(fact_8543_ATP_Olambda__424,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bj(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_424
tff(fact_8544_ATP_Olambda__425,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ip(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_425
tff(fact_8545_ATP_Olambda__426,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_is(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_426
tff(fact_8546_ATP_Olambda__427,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_kj(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(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),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_427
tff(fact_8547_ATP_Olambda__428,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_kx(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(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),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).

% ATP.lambda_428
tff(fact_8548_ATP_Olambda__429,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_ki(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(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),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_429
tff(fact_8549_ATP_Olambda__430,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,bool),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_it(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(nat,bool,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_430
tff(fact_8550_ATP_Olambda__431,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_jf(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_431
tff(fact_8551_ATP_Olambda__432,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_add(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_432
tff(fact_8552_ATP_Olambda__433,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_jg(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_433
tff(fact_8553_ATP_Olambda__434,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,option(A),aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_adq(fun(B,A),fun(fun(B,bool),fun(B,option(A))),Uu),Uua),Uub) = if(option(A),aa(B,bool,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_434
tff(fact_8554_ATP_Olambda__435,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_agw(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_agv(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uub)),Uua) ).

% ATP.lambda_435
tff(fact_8555_ATP_Olambda__436,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( pp(aa(real,bool,aa(fun(real,real),fun(real,bool),aTP_Lamp_xm(fun(real,real),fun(fun(real,real),fun(real,bool)),Uu),Uua),Uub))
    <=> has_field_derivative(real,Uu,aa(real,real,Uua,Uub),topolo174197925503356063within(real,Uub,top_top(set(real)))) ) ).

% ATP.lambda_436
tff(fact_8556_ATP_Olambda__437,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_abc(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_437
tff(fact_8557_ATP_Olambda__438,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: option(B),Uub: A] :
      ( pp(aa(A,bool,aa(option(B),fun(A,bool),aTP_Lamp_agz(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),aa(bool,fun(fun(B,bool),fun(option(B),bool)),case_option(bool,B),fFalse),aa(A,fun(B,bool),Uu,Uub)),Uua)) ) ).

% ATP.lambda_438
tff(fact_8558_ATP_Olambda__439,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_aat(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,aa(B,C,Uua,Uub)),Uub) ) ).

% ATP.lambda_439
tff(fact_8559_ATP_Olambda__440,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_hw(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_440
tff(fact_8560_ATP_Olambda__441,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_hu(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_441
tff(fact_8561_ATP_Olambda__442,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_qa(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu,Uub),Uua) ).

% ATP.lambda_442
tff(fact_8562_ATP_Olambda__443,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_eu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_443
tff(fact_8563_ATP_Olambda__444,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_er(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_444
tff(fact_8564_ATP_Olambda__445,axiom,
    ! [I6: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(I6,fun(A,B)),Uua: A,Uub: I6] : aa(I6,B,aa(A,fun(I6,B),aTP_Lamp_rq(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uu),Uua),Uub) = aa(A,B,aa(I6,fun(A,B),Uu,Uub),Uua) ) ).

% ATP.lambda_445
tff(fact_8565_ATP_Olambda__446,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_aar(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_446
tff(fact_8566_ATP_Olambda__447,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_tw(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_447
tff(fact_8567_ATP_Olambda__448,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_tu(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_448
tff(fact_8568_ATP_Olambda__449,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_go(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_449
tff(fact_8569_ATP_Olambda__450,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_gm(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gl(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_450
tff(fact_8570_ATP_Olambda__451,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_gk(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gj(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_451
tff(fact_8571_ATP_Olambda__452,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_fh(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fg(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_452
tff(fact_8572_ATP_Olambda__453,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_co(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uub),Uu),one_one(A),zero_zero(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_453
tff(fact_8573_ATP_Olambda__454,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,bool),aa(code_integer,fun(code_integer,product_prod(code_integer,bool)),aTP_Lamp_kh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),Uu),Uua),Uub) = aa(bool,product_prod(code_integer,bool),product_Pair(code_integer,bool,if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),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))),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),one_one(code_integer))) ).

% ATP.lambda_454
tff(fact_8574_ATP_Olambda__455,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_ev(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_eu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_455
tff(fact_8575_ATP_Olambda__456,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_es(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_er(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_456
tff(fact_8576_ATP_Olambda__457,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_ut(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_457
tff(fact_8577_ATP_Olambda__458,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kw(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_kv(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_458
tff(fact_8578_ATP_Olambda__459,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ku(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_kt(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_459
tff(fact_8579_ATP_Olambda__460,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ks(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aa(int,fun(int,fun(int,bool)),aTP_Lamp_kr(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_460
tff(fact_8580_ATP_Olambda__461,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: A] : aa(A,B,aa(fun(I6,fun(A,B)),fun(A,B),aTP_Lamp_rr(set(I6),fun(fun(I6,fun(A,B)),fun(A,B)),Uu),Uua),Uub) = aa(set(I6),B,aa(fun(I6,B),fun(set(I6),B),groups7121269368397514597t_prod(I6,B),aa(A,fun(I6,B),aTP_Lamp_rq(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_461
tff(fact_8581_ATP_Olambda__462,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qg(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_462
tff(fact_8582_ATP_Olambda__463,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_qh(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_463
tff(fact_8583_ATP_Olambda__464,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_dz(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_464
tff(fact_8584_ATP_Olambda__465,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_jv(fun(nat,A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_465
tff(fact_8585_ATP_Olambda__466,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_um(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_466
tff(fact_8586_ATP_Olambda__467,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_467
tff(fact_8587_ATP_Olambda__468,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_iw(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_468
tff(fact_8588_ATP_Olambda__469,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_sk(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_469
tff(fact_8589_ATP_Olambda__470,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_470
tff(fact_8590_ATP_Olambda__471,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qd(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_471
tff(fact_8591_ATP_Olambda__472,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ea(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_472
tff(fact_8592_ATP_Olambda__473,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_on(nat,fun(nat,fun(list(nat),bool)),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_473
tff(fact_8593_ATP_Olambda__474,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_ei(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_474
tff(fact_8594_ATP_Olambda__475,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_nz(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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) )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Uua),Uub)),lex(A,Uu))) ) ) ) ).

% ATP.lambda_475
tff(fact_8595_ATP_Olambda__476,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_ahl(nat,fun(list(A),fun(list(A),bool)),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_476
tff(fact_8596_ATP_Olambda__477,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_acm(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X4: A,Y4: 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),X4),Xs5)) )
            & ( Uub = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X4),Y4)),Uu)) ) ) ) ).

% ATP.lambda_477
tff(fact_8597_ATP_Olambda__478,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_oo(nat,fun(nat,fun(list(nat),bool)),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_478
tff(fact_8598_ATP_Olambda__479,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(A),fun(list(A),bool),aTP_Lamp_ll(nat,fun(set(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua)) ) ) ).

% ATP.lambda_479
tff(fact_8599_ATP_Olambda__480,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_lk(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)) ) ) ).

% ATP.lambda_480
tff(fact_8600_ATP_Olambda__481,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_afk(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_481
tff(fact_8601_ATP_Olambda__482,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_kd(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua))) ) ) ).

% ATP.lambda_482
tff(fact_8602_ATP_Olambda__483,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_iz(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua)) ) ) ).

% ATP.lambda_483
tff(fact_8603_ATP_Olambda__484,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_iy(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_484
tff(fact_8604_ATP_Olambda__485,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_om(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_485
tff(fact_8605_ATP_Olambda__486,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_px(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_486
tff(fact_8606_ATP_Olambda__487,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_le(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_487
tff(fact_8607_ATP_Olambda__488,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_adm(set(nat),fun(nat,fun(product_prod(A,nat),bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),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_488
tff(fact_8608_ATP_Olambda__489,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),aTP_Lamp_afw(nat,fun(nat,fun(set(nat),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(nat)),bool,aa(set(nat),fun(set(set(nat)),bool),member(set(nat)),Uub),pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu))))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_489
tff(fact_8609_ATP_Olambda__490,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_op(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)))
        & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uua)) ) ) ).

% ATP.lambda_490
tff(fact_8610_ATP_Olambda__491,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_adf(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))
        & pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ) ).

% ATP.lambda_491
tff(fact_8611_ATP_Olambda__492,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(A,fun(product_prod(A,B),bool),aTP_Lamp_acr(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),Uub),graph(A,B,Uu)))
        & ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).

% ATP.lambda_492
tff(fact_8612_ATP_Olambda__493,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ig(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),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_493
tff(fact_8613_ATP_Olambda__494,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_aco(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),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_494
tff(fact_8614_ATP_Olambda__495,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_agm(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uub)) ) ) ).

% ATP.lambda_495
tff(fact_8615_ATP_Olambda__496,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_aeb(set(A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,Uu,Uua)),Uub)) ) ) ) ).

% ATP.lambda_496
tff(fact_8616_ATP_Olambda__497,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lf(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(nat,nat,suc,Uua))) ) ) ).

% ATP.lambda_497
tff(fact_8617_ATP_Olambda__498,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_eh(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_498
tff(fact_8618_ATP_Olambda__499,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_ez(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_499
tff(fact_8619_ATP_Olambda__500,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_fr(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_500
tff(fact_8620_ATP_Olambda__501,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fc(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_501
tff(fact_8621_ATP_Olambda__502,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_kc(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uua))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_502
tff(fact_8622_ATP_Olambda__503,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_jw(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_503
tff(fact_8623_ATP_Olambda__504,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_kb(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_504
tff(fact_8624_ATP_Olambda__505,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_jy(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_505
tff(fact_8625_ATP_Olambda__506,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_jz(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_506
tff(fact_8626_ATP_Olambda__507,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_jx(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_507
tff(fact_8627_ATP_Olambda__508,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_adw(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uu)) ) ) ).

% ATP.lambda_508
tff(fact_8628_ATP_Olambda__509,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_adu(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uub),Uua))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uub),Uu)) ) ) ).

% ATP.lambda_509
tff(fact_8629_ATP_Olambda__510,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_je(set(B),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_510
tff(fact_8630_ATP_Olambda__511,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_hj(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_511
tff(fact_8631_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_512
tff(fact_8632_ATP_Olambda__513,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ke(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_513
tff(fact_8633_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_514
tff(fact_8634_ATP_Olambda__515,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_kf(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua))
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_515
tff(fact_8635_ATP_Olambda__516,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_mc(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua))
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_516
tff(fact_8636_ATP_Olambda__517,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aj(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uua)) ) ) ).

% ATP.lambda_517
tff(fact_8637_ATP_Olambda__518,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_adv(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uu)),Uua)) ) ).

% ATP.lambda_518
tff(fact_8638_ATP_Olambda__519,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ht(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_519
tff(fact_8639_ATP_Olambda__520,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_hy(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_520
tff(fact_8640_ATP_Olambda__521,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mv(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_521
tff(fact_8641_ATP_Olambda__522,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mt(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_522
tff(fact_8642_ATP_Olambda__523,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mu(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_523
tff(fact_8643_ATP_Olambda__524,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ms(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_524
tff(fact_8644_ATP_Olambda__525,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_oh(set(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,Uua),Uub)),Uu)) ) ).

% ATP.lambda_525
tff(fact_8645_ATP_Olambda__526,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_acw(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_526
tff(fact_8646_ATP_Olambda__527,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa(complex,bool,aa(complex,fun(complex,bool),aTP_Lamp_di(nat,fun(complex,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu) = Uua ) ) ).

% ATP.lambda_527
tff(fact_8647_ATP_Olambda__528,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa(complex,bool,aa(nat,fun(complex,bool),aTP_Lamp_bi(complex,fun(nat,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu ) ) ).

% ATP.lambda_528
tff(fact_8648_ATP_Olambda__529,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_jh(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uub))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uub),Uua)) ) ) ) ).

% ATP.lambda_529
tff(fact_8649_ATP_Olambda__530,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bx(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_530
tff(fact_8650_ATP_Olambda__531,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_bw(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_531
tff(fact_8651_ATP_Olambda__532,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_we(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_532
tff(fact_8652_ATP_Olambda__533,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qe(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_533
tff(fact_8653_ATP_Olambda__534,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_bz(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_534
tff(fact_8654_ATP_Olambda__535,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fo(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_535
tff(fact_8655_ATP_Olambda__536,axiom,
    ! [Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(nat,Aa),Uua: Aa,Uub: nat] : aa(nat,Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_uu(fun(nat,Aa),fun(Aa,fun(nat,Aa)),Uu),Uua),Uub) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uu,Uub)),aa(nat,Aa,aa(Aa,fun(nat,Aa),power_power(Aa),Uua),Uub)) ) ).

% ATP.lambda_536
tff(fact_8656_ATP_Olambda__537,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_ep(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_537
tff(fact_8657_ATP_Olambda__538,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_br(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_538
tff(fact_8658_ATP_Olambda__539,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_bv(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_539
tff(fact_8659_ATP_Olambda__540,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_ey(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_540
tff(fact_8660_ATP_Olambda__541,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_fa(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_541
tff(fact_8661_ATP_Olambda__542,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fe(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_542
tff(fact_8662_ATP_Olambda__543,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ix(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,Uu,Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_543
tff(fact_8663_ATP_Olambda__544,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uu,Uub)),aa(real,real,Uua,Uub)) ).

% ATP.lambda_544
tff(fact_8664_ATP_Olambda__545,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ro(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(C,A,Uu,Uub)),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_545
tff(fact_8665_ATP_Olambda__546,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_rd(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(C,A,Uu,Uub)),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_546
tff(fact_8666_ATP_Olambda__547,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_xx(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(C,A,Uu,Uub)),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_547
tff(fact_8667_ATP_Olambda__548,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tm(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_548
tff(fact_8668_ATP_Olambda__549,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_bg(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_549
tff(fact_8669_ATP_Olambda__550,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_yw(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_550
tff(fact_8670_ATP_Olambda__551,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_zb(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_551
tff(fact_8671_ATP_Olambda__552,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,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_552
tff(fact_8672_ATP_Olambda__553,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_ue(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_553
tff(fact_8673_ATP_Olambda__554,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_xr(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_554
tff(fact_8674_ATP_Olambda__555,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xp(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uua,Uub)),aa(real,real,Uu,Uub)) ).

% ATP.lambda_555
tff(fact_8675_ATP_Olambda__556,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_zm(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ) ).

% ATP.lambda_556
tff(fact_8676_ATP_Olambda__557,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_oq(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_557
tff(fact_8677_ATP_Olambda__558,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_il(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_558
tff(fact_8678_ATP_Olambda__559,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_ik(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_559
tff(fact_8679_ATP_Olambda__560,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_tk(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_560
tff(fact_8680_ATP_Olambda__561,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D] : aa(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_tp(fun(D,A),fun(fun(D,A),fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),aa(D,A,Uua,Uub)) ) ).

% ATP.lambda_561
tff(fact_8681_ATP_Olambda__562,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xv(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_562
tff(fact_8682_ATP_Olambda__563,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_md(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_563
tff(fact_8683_ATP_Olambda__564,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_cr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_564
tff(fact_8684_ATP_Olambda__565,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_vk(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_565
tff(fact_8685_ATP_Olambda__566,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ox(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_566
tff(fact_8686_ATP_Olambda__567,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_567
tff(fact_8687_ATP_Olambda__568,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_agt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_568
tff(fact_8688_ATP_Olambda__569,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_zc(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_569
tff(fact_8689_ATP_Olambda__570,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tq(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_570
tff(fact_8690_ATP_Olambda__571,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tj(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_571
tff(fact_8691_ATP_Olambda__572,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cf(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_572
tff(fact_8692_ATP_Olambda__573,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_qt(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_573
tff(fact_8693_ATP_Olambda__574,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_pl(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_574
tff(fact_8694_ATP_Olambda__575,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agh(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_575
tff(fact_8695_ATP_Olambda__576,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_uf(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_576
tff(fact_8696_ATP_Olambda__577,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ti(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_577
tff(fact_8697_ATP_Olambda__578,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_afj(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_578
tff(fact_8698_ATP_Olambda__579,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_aht(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_579
tff(fact_8699_ATP_Olambda__580,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_jp(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_580
tff(fact_8700_ATP_Olambda__581,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ts(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uua,Uub)),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_581
tff(fact_8701_ATP_Olambda__582,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_dq(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_582
tff(fact_8702_ATP_Olambda__583,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_dg(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_583
tff(fact_8703_ATP_Olambda__584,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_ow(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_584
tff(fact_8704_ATP_Olambda__585,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_sm(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_585
tff(fact_8705_ATP_Olambda__586,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_kg(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_586
tff(fact_8706_ATP_Olambda__587,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_pz(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,Uu,Uub)),aa(real,real,Uua,Uub)) ).

% ATP.lambda_587
tff(fact_8707_ATP_Olambda__588,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [Uu: fun(C,real),Uua: fun(C,real),Uub: C] : aa(C,real,aa(fun(C,real),fun(C,real),aTP_Lamp_zl(fun(C,real),fun(fun(C,real),fun(C,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),powr(real),aa(C,real,Uu,Uub)),aa(C,real,Uua,Uub)) ) ).

% ATP.lambda_588
tff(fact_8708_ATP_Olambda__589,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [Uu: fun(C,real),Uua: fun(C,real),Uub: C] : aa(C,real,aa(fun(C,real),fun(C,real),aTP_Lamp_uk(fun(C,real),fun(fun(C,real),fun(C,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),powr(real),aa(C,real,Uu,Uub)),aa(C,real,Uua,Uub)) ) ).

% ATP.lambda_589
tff(fact_8709_ATP_Olambda__590,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_ru(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),powr(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_590
tff(fact_8710_ATP_Olambda__591,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_xt(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),powr(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_591
tff(fact_8711_ATP_Olambda__592,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_sr(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),powr(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_592
tff(fact_8712_ATP_Olambda__593,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_zp(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log2(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_593
tff(fact_8713_ATP_Olambda__594,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_ur(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log2(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_594
tff(fact_8714_ATP_Olambda__595,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,log2(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_595
tff(fact_8715_ATP_Olambda__596,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_abv(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_596
tff(fact_8716_ATP_Olambda__597,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ahi(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_597
tff(fact_8717_ATP_Olambda__598,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_xd(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua)))) ) ) ).

% ATP.lambda_598
tff(fact_8718_ATP_Olambda__599,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_pr(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_599
tff(fact_8719_ATP_Olambda__600,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(C,A),Uua: nat,Uub: C] : aa(C,A,aa(nat,fun(C,A),aTP_Lamp_zi(fun(C,A),fun(nat,fun(C,A)),Uu),Uua),Uub) = comm_s3205402744901411588hammer(A,aa(C,A,Uu,Uub),Uua) ) ).

% ATP.lambda_600
tff(fact_8720_ATP_Olambda__601,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,set(A),aa(A,fun(B,set(A)),aTP_Lamp_afd(fun(B,A),fun(A,fun(B,set(A))),Uu),Uua),Uub) = set_or3652927894154168847AtMost(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_601
tff(fact_8721_ATP_Olambda__602,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_tx(set(B),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_602
tff(fact_8722_ATP_Olambda__603,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_tv(set(B),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_603
tff(fact_8723_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_wu(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_604
tff(fact_8724_ATP_Olambda__605,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ys(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_605
tff(fact_8725_ATP_Olambda__606,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_bm(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_606
tff(fact_8726_ATP_Olambda__607,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_mo(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_607
tff(fact_8727_ATP_Olambda__608,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_tl(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_608
tff(fact_8728_ATP_Olambda__609,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_cg(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_609
tff(fact_8729_ATP_Olambda__610,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_pk(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_610
tff(fact_8730_ATP_Olambda__611,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_va(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_611
tff(fact_8731_ATP_Olambda__612,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_wq(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_612
tff(fact_8732_ATP_Olambda__613,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_wx(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_613
tff(fact_8733_ATP_Olambda__614,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_yq(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_614
tff(fact_8734_ATP_Olambda__615,axiom,
    ! [D: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D,A),Uua: A,Uub: D] : aa(D,A,aa(A,fun(D,A),aTP_Lamp_to(fun(D,A),fun(A,fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_615
tff(fact_8735_ATP_Olambda__616,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_yv(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_616
tff(fact_8736_ATP_Olambda__617,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_ci(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_617
tff(fact_8737_ATP_Olambda__618,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_uz(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_618
tff(fact_8738_ATP_Olambda__619,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_sc(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_619
tff(fact_8739_ATP_Olambda__620,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ags(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_620
tff(fact_8740_ATP_Olambda__621,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dr(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_621
tff(fact_8741_ATP_Olambda__622,axiom,
    ! [K8: $tType,L5: $tType,Uu: fun(K8,set(L5)),Uua: set(L5),Uub: K8] : aa(K8,set(L5),aa(set(L5),fun(K8,set(L5)),aTP_Lamp_aaf(fun(K8,set(L5)),fun(set(L5),fun(K8,set(L5))),Uu),Uua),Uub) = aa(set(L5),set(L5),aa(set(L5),fun(set(L5),set(L5)),minus_minus(set(L5)),aa(K8,set(L5),Uu,Uub)),Uua) ).

% ATP.lambda_622
tff(fact_8742_ATP_Olambda__623,axiom,
    ! [E5: $tType,F: $tType,Uu: fun(E5,set(F)),Uua: set(F),Uub: E5] : aa(E5,set(F),aa(set(F),fun(E5,set(F)),aTP_Lamp_aaj(fun(E5,set(F)),fun(set(F),fun(E5,set(F))),Uu),Uua),Uub) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(E5,set(F),Uu,Uub)),Uua) ).

% ATP.lambda_623
tff(fact_8743_ATP_Olambda__624,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_tr(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_624
tff(fact_8744_ATP_Olambda__625,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_pv(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_625
tff(fact_8745_ATP_Olambda__626,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_rk(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_626
tff(fact_8746_ATP_Olambda__627,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_ps(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_627
tff(fact_8747_ATP_Olambda__628,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_xy(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_628
tff(fact_8748_ATP_Olambda__629,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_629
tff(fact_8749_ATP_Olambda__630,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yj(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_630
tff(fact_8750_ATP_Olambda__631,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_se(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_631
tff(fact_8751_ATP_Olambda__632,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_abw(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_632
tff(fact_8752_ATP_Olambda__633,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_zz(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_633
tff(fact_8753_ATP_Olambda__634,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_py(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),powr(real),aa(real,real,Uu,Uub)),Uua) ).

% ATP.lambda_634
tff(fact_8754_ATP_Olambda__635,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xu(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),powr(real),aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_635
tff(fact_8755_ATP_Olambda__636,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,A),Uua: int,Uub: C] : aa(C,A,aa(int,fun(C,A),aTP_Lamp_abj(fun(C,A),fun(int,fun(C,A)),Uu),Uua),Uub) = power_int(A,aa(C,A,Uu,Uub),Uua) ) ).

% ATP.lambda_636
tff(fact_8756_ATP_Olambda__637,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(B,A),Uua: int,Uub: B] : aa(B,A,aa(int,fun(B,A),aTP_Lamp_abm(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_637
tff(fact_8757_ATP_Olambda__638,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_abo(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_638
tff(fact_8758_ATP_Olambda__639,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_abl(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_639
tff(fact_8759_ATP_Olambda__640,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_abp(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_640
tff(fact_8760_ATP_Olambda__641,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_abn(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_641
tff(fact_8761_ATP_Olambda__642,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_agu(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_642
tff(fact_8762_ATP_Olambda__643,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_agp(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(B,A,Uu,Uub) = Uua ) ) ).

% ATP.lambda_643
tff(fact_8763_ATP_Olambda__644,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_if(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_644
tff(fact_8764_ATP_Olambda__645,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_acj(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_645
tff(fact_8765_ATP_Olambda__646,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_ha(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_646
tff(fact_8766_ATP_Olambda__647,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_xc(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_647
tff(fact_8767_ATP_Olambda__648,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),bool)),Uua: list(A),Uub: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_acp(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub))
        <=> ( ? [Y4: A,Ys3: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) ) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys3: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4)) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys3: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X4))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uu,Xs3),Ys3)) ) ) ) ) ).

% ATP.lambda_648
tff(fact_8768_ATP_Olambda__649,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(bool,fun(bool,A)),Uub: vEBT_VEBT] : aa(vEBT_VEBT,product_prod(vEBT_VEBT,A),aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_aeh(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),Uu),Uua),Uub) = aa(A,product_prod(vEBT_VEBT,A),product_Pair(vEBT_VEBT,A,Uub),aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),Uu),Uua),Uub)) ).

% ATP.lambda_649
tff(fact_8769_ATP_Olambda__650,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_id(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(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_650
tff(fact_8770_ATP_Olambda__651,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_abi(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_651
tff(fact_8771_ATP_Olambda__652,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dc(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_652
tff(fact_8772_ATP_Olambda__653,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_da(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_653
tff(fact_8773_ATP_Olambda__654,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aeg(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),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,Uua),bot_bot(set(A)))))) ) ).

% ATP.lambda_654
tff(fact_8774_ATP_Olambda__655,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_dy(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_655
tff(fact_8775_ATP_Olambda__656,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ack(fun(A,option(B)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),dom(A,B,Uu)))) ) ).

% ATP.lambda_656
tff(fact_8776_ATP_Olambda__657,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_acl(fun(A,option(B)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),dom(A,B,Uu)))) ) ).

% ATP.lambda_657
tff(fact_8777_ATP_Olambda__658,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_vs(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_658
tff(fact_8778_ATP_Olambda__659,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_db(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_659
tff(fact_8779_ATP_Olambda__660,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: real,Uub: A] : aa(A,A,aa(real,fun(A,A),aTP_Lamp_pj(fun(A,A),fun(real,fun(A,A)),Uu),Uua),Uub) = real_V8093663219630862766scaleR(A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_660
tff(fact_8780_ATP_Olambda__661,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_wt(A,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_661
tff(fact_8781_ATP_Olambda__662,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_wo(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_662
tff(fact_8782_ATP_Olambda__663,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_ky(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_663
tff(fact_8783_ATP_Olambda__664,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_wp(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_664
tff(fact_8784_ATP_Olambda__665,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_adr(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_665
tff(fact_8785_ATP_Olambda__666,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_wm(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_666
tff(fact_8786_ATP_Olambda__667,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_yr(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_667
tff(fact_8787_ATP_Olambda__668,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_bl(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_668
tff(fact_8788_ATP_Olambda__669,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yu(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_669
tff(fact_8789_ATP_Olambda__670,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_cj(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_670
tff(fact_8790_ATP_Olambda__671,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_uy(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_671
tff(fact_8791_ATP_Olambda__672,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_sd(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_672
tff(fact_8792_ATP_Olambda__673,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D,A),Uua: A,Uub: D] : aa(D,A,aa(A,fun(D,A),aTP_Lamp_tn(fun(D,A),fun(A,fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(D,A,Uu,Uub)) ) ).

% ATP.lambda_673
tff(fact_8793_ATP_Olambda__674,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_yp(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_674
tff(fact_8794_ATP_Olambda__675,axiom,
    ! [M11: $tType,N10: $tType,Uu: set(M11),Uua: fun(N10,set(M11)),Uub: N10] : aa(N10,set(M11),aa(fun(N10,set(M11)),fun(N10,set(M11)),aTP_Lamp_aan(set(M11),fun(fun(N10,set(M11)),fun(N10,set(M11))),Uu),Uua),Uub) = aa(set(M11),set(M11),aa(set(M11),fun(set(M11),set(M11)),minus_minus(set(M11)),Uu),aa(N10,set(M11),Uua,Uub)) ).

% ATP.lambda_675
tff(fact_8795_ATP_Olambda__676,axiom,
    ! [G4: $tType,H6: $tType,Uu: set(G4),Uua: fun(H6,set(G4)),Uub: H6] : aa(H6,set(G4),aa(fun(H6,set(G4)),fun(H6,set(G4)),aTP_Lamp_aak(set(G4),fun(fun(H6,set(G4)),fun(H6,set(G4))),Uu),Uua),Uub) = aa(set(G4),set(G4),aa(set(G4),fun(set(G4),set(G4)),minus_minus(set(G4)),Uu),aa(H6,set(G4),Uua,Uub)) ).

% ATP.lambda_676
tff(fact_8796_ATP_Olambda__677,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(B,nat),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_vy(fun(B,nat),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(B,nat,Uu,Uub)) ) ).

% ATP.lambda_677
tff(fact_8797_ATP_Olambda__678,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aby(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_678
tff(fact_8798_ATP_Olambda__679,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aab(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_679
tff(fact_8799_ATP_Olambda__680,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_mp(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_680
tff(fact_8800_ATP_Olambda__681,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ax(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_681
tff(fact_8801_ATP_Olambda__682,axiom,
    ! [Uu: real,Uua: real,Uub: product_unit] : aa(product_unit,real,aa(real,fun(product_unit,real),aTP_Lamp_afa(real,fun(real,fun(product_unit,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),powr_real,Uu),Uua) ).

% ATP.lambda_682
tff(fact_8802_ATP_Olambda__683,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: fun(A,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aTP_Lamp_agq(fun(A,B),fun(fun(A,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,Uua,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),Uu),Uub))) ) ).

% ATP.lambda_683
tff(fact_8803_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_dp(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_684
tff(fact_8804_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_do(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_685
tff(fact_8805_ATP_Olambda__686,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vh(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_686
tff(fact_8806_ATP_Olambda__687,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vc(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uua)) ) ).

% ATP.lambda_687
tff(fact_8807_ATP_Olambda__688,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_sj(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_688
tff(fact_8808_ATP_Olambda__689,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dt(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_689
tff(fact_8809_ATP_Olambda__690,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,bool),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ws(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua))) ) ) ).

% ATP.lambda_690
tff(fact_8810_ATP_Olambda__691,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_sg(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_691
tff(fact_8811_ATP_Olambda__692,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_uo(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_692
tff(fact_8812_ATP_Olambda__693,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),bool),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_in(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(A,B),bool,Uu,aa(B,product_prod(A,B),product_Pair(A,B,Uua),Uub))) ) ).

% ATP.lambda_693
tff(fact_8813_ATP_Olambda__694,axiom,
    ! [D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D) )
     => ! [Uu: A,Uua: fun(A,D),Uub: A] : aa(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_ub(A,fun(fun(A,D),fun(A,D)),Uu),Uua),Uub) = aa(A,D,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_694
tff(fact_8814_ATP_Olambda__695,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_si(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_695
tff(fact_8815_ATP_Olambda__696,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_cm(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_696
tff(fact_8816_ATP_Olambda__697,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(B,bool)),Uua: fun(A,C),Uub: A] : aa(A,fun(B,bool),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ahg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Uu),Uua),Uub) = aa(C,fun(B,bool),Uu,aa(A,C,Uua,Uub)) ).

% ATP.lambda_697
tff(fact_8817_ATP_Olambda__698,axiom,
    ! [C: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(B,C),Uua: fun(nat,B),Uub: nat] : aa(nat,C,aa(fun(nat,B),fun(nat,C),aTP_Lamp_yy(fun(B,C),fun(fun(nat,B),fun(nat,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(nat,B,Uua,Uub)) ) ).

% ATP.lambda_698
tff(fact_8818_ATP_Olambda__699,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_zy(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ) ).

% ATP.lambda_699
tff(fact_8819_ATP_Olambda__700,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,B),Uub: A] : aa(A,C,aa(fun(A,B),fun(A,C),aTP_Lamp_na(fun(B,C),fun(fun(A,B),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(A,B,Uua,Uub)) ).

% ATP.lambda_700
tff(fact_8820_ATP_Olambda__701,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,bool),Uua: fun(nat,A),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_yt(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(nat,A,Uua,Uub))) ) ) ).

% ATP.lambda_701
tff(fact_8821_ATP_Olambda__702,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wr(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_702
tff(fact_8822_ATP_Olambda__703,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(num,A),Uub: num] : aa(num,B,aa(fun(num,A),fun(num,B),aTP_Lamp_no(fun(A,B),fun(fun(num,A),fun(num,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(num,A,Uua,Uub)) ).

% ATP.lambda_703
tff(fact_8823_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ae(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_8824_ATP_Olambda__705,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,nat),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aet(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_705
tff(fact_8825_ATP_Olambda__706,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,nat),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aes(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_706
tff(fact_8826_ATP_Olambda__707,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_jr(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_707
tff(fact_8827_ATP_Olambda__708,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_jo(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_708
tff(fact_8828_ATP_Olambda__709,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_sf(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_8829_ATP_Olambda__710,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_mw(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_8830_ATP_Olambda__711,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_aho(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = product_case_prod(A,A,bool,aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_ahn(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ).

% ATP.lambda_711
tff(fact_8831_ATP_Olambda__712,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A] : aa(A,Aa,aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_uw(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),Uu),Uua),Uub) = suminf(Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_uv(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub)) ) ).

% ATP.lambda_712
tff(fact_8832_ATP_Olambda__713,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_sb(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_sa(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_713
tff(fact_8833_ATP_Olambda__714,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_ahy(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),Uu),Uua),Uub) = 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),Uua),aa(A,fun(A,A),Uu,Uua)),Uub)) ).

% ATP.lambda_714
tff(fact_8834_ATP_Olambda__715,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_agv(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_715
tff(fact_8835_ATP_Olambda__716,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_agk(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( real_V7696804695334737415tation(A,real_V4986007116245087402_basis(A,Uu),Uua,Uub) != zero_zero(real) ) ) ) ).

% ATP.lambda_716
tff(fact_8836_ATP_Olambda__717,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: set(A),Uub: A] : aa(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_abb(A,fun(set(A),fun(A,filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_greaterThan(A,Uub)),Uua)),aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_717
tff(fact_8837_ATP_Olambda__718,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: set(A),Uub: A] : aa(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aba(A,fun(set(A),fun(A,filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,Uub)),Uua)),aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_718
tff(fact_8838_ATP_Olambda__719,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: set(A),Uub: set(A)] : aa(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_abf(A,fun(set(A),fun(set(A),filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uub),Uua)),aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_719
tff(fact_8839_ATP_Olambda__720,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_bs(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_720
tff(fact_8840_ATP_Olambda__721,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(I6,A),Uua: fun(I6,A),Uub: I6] : aa(I6,real,aa(fun(I6,A),fun(I6,real),aTP_Lamp_dj(fun(I6,A),fun(fun(I6,A),fun(I6,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(I6,A,Uu,Uub)),aa(I6,A,Uua,Uub))) ) ).

% ATP.lambda_721
tff(fact_8841_ATP_Olambda__722,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_wd(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_722
tff(fact_8842_ATP_Olambda__723,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_mq(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_723
tff(fact_8843_ATP_Olambda__724,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_ob(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [I2: nat] :
          ( ( Uub = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Uu),I2)),aa(nat,B,nth(B,Uua),I2)) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),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_724
tff(fact_8844_ATP_Olambda__725,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_ol(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [I2: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I2) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Uu)))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),I2),Uua)) ) ) ).

% ATP.lambda_725
tff(fact_8845_ATP_Olambda__726,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A,Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_age(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [K3: real] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),real_V8093663219630862766scaleR(A,K3,Uu))),real_Vector_span(A,Uua))) ) ) ).

% ATP.lambda_726
tff(fact_8846_ATP_Olambda__727,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_wj(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [A5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uub),A5))
             => ! [B3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A5),B3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or3652927894154168847AtMost(nat,A5,B3)))),aa(nat,real,Uua,A5))) ) ) ) ) ).

% ATP.lambda_727
tff(fact_8847_ATP_Olambda__728,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aaz(fun(A,A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ? [N5: 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)),N5),Uu),Uua) ) ).

% ATP.lambda_728
tff(fact_8848_ATP_Olambda__729,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_gn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uuc)),aa(A,A,aa(A,fun(A,A),times_times(A),real_V8093663219630862766scaleR(A,aa(real,real,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_729
tff(fact_8849_ATP_Olambda__730,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_gj(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uuc))),aa(A,A,aa(A,fun(A,A),times_times(A),real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),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_730
tff(fact_8850_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_gl(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),real_V8093663219630862766scaleR(A,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_8851_ATP_Olambda__732,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_fs(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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_732
tff(fact_8852_ATP_Olambda__733,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_fu(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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_733
tff(fact_8853_ATP_Olambda__734,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_fv(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_734
tff(fact_8854_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_ft(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_735
tff(fact_8855_ATP_Olambda__736,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_jq(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uuc),Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_736
tff(fact_8856_ATP_Olambda__737,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ju(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_737
tff(fact_8857_ATP_Olambda__738,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_jt(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_738
tff(fact_8858_ATP_Olambda__739,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_li(B,fun(fun(B,A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),Uub) ) ).

% ATP.lambda_739
tff(fact_8859_ATP_Olambda__740,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: B,Uub: set(A),Uuc: A] : aa(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ael(B,fun(B,fun(set(A),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uub),Uu,Uua) ).

% ATP.lambda_740
tff(fact_8860_ATP_Olambda__741,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mn(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_741
tff(fact_8861_ATP_Olambda__742,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mm(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_742
tff(fact_8862_ATP_Olambda__743,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_743
tff(fact_8863_ATP_Olambda__744,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( 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_aia(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_744
tff(fact_8864_ATP_Olambda__745,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: B,Uuc: D] : aa(D,C,aa(B,fun(D,C),aa(fun(D,A),fun(B,fun(D,C)),aTP_Lamp_aib(fun(A,fun(B,C)),fun(fun(D,A),fun(B,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),Uub) ) ).

% ATP.lambda_745
tff(fact_8865_ATP_Olambda__746,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_ahz(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_746
tff(fact_8866_ATP_Olambda__747,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,C),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ahf(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),aa(B,C,Uua,Uuc))) ) ).

% ATP.lambda_747
tff(fact_8867_ATP_Olambda__748,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_eg(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ef(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_748
tff(fact_8868_ATP_Olambda__749,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_fy(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),zero_zero(nat)),aa(A,A,uminus_uminus(A),Uub),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),zero_zero(A)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_749
tff(fact_8869_ATP_Olambda__750,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: set(A),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_agj(set(A),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = real_V8093663219630862766scaleR(B,real_V7696804695334737415tation(A,real_V4986007116245087402_basis(A,Uu),Uub,Uuc),if(B,aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu),aa(A,B,Uua,Uuc),zero_zero(B))) ) ).

% ATP.lambda_750
tff(fact_8870_ATP_Olambda__751,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_fq(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,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fp(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_751
tff(fact_8871_ATP_Olambda__752,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_fk(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fj(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_752
tff(fact_8872_ATP_Olambda__753,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_qm(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_753
tff(fact_8873_ATP_Olambda__754,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_qk(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_754
tff(fact_8874_ATP_Olambda__755,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_qi(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_755
tff(fact_8875_ATP_Olambda__756,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_qj(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_756
tff(fact_8876_ATP_Olambda__757,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_du(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_757
tff(fact_8877_ATP_Olambda__758,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_lo(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(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_758
tff(fact_8878_ATP_Olambda__759,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_ef(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_759
tff(fact_8879_ATP_Olambda__760,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_wa(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_760
tff(fact_8880_ATP_Olambda__761,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_aen(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Uub),Uuc)),Uua)) ) ) ) ).

% ATP.lambda_761
tff(fact_8881_ATP_Olambda__762,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_fn(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_762
tff(fact_8882_ATP_Olambda__763,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_fi(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_763
tff(fact_8883_ATP_Olambda__764,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_fm(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_764
tff(fact_8884_ATP_Olambda__765,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_gy(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_765
tff(fact_8885_ATP_Olambda__766,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_acq(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X4: A,Y4: 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),X4),Xs5)) )
            & ( Uuc = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X4),Y4)),Uu)) ) ) ) ).

% ATP.lambda_766
tff(fact_8886_ATP_Olambda__767,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_dw(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_767
tff(fact_8887_ATP_Olambda__768,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_dv(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_768
tff(fact_8888_ATP_Olambda__769,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_mx(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_769
tff(fact_8889_ATP_Olambda__770,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_dx(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_770
tff(fact_8890_ATP_Olambda__771,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_fd(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_771
tff(fact_8891_ATP_Olambda__772,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_lu(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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_772
tff(fact_8892_ATP_Olambda__773,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_kr(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).

% ATP.lambda_773
tff(fact_8893_ATP_Olambda__774,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_ls(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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_774
tff(fact_8894_ATP_Olambda__775,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_lw(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(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_775
tff(fact_8895_ATP_Olambda__776,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_ly(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(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_776
tff(fact_8896_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_im(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_777
tff(fact_8897_ATP_Olambda__778,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_acd(nat,fun(nat,fun(nat,fun(nat,bool))),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_778
tff(fact_8898_ATP_Olambda__779,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_jd(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != one_one(A) ) ) ) ) ).

% ATP.lambda_779
tff(fact_8899_ATP_Olambda__780,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_jb(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_780
tff(fact_8900_ATP_Olambda__781,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_wk(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = 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_781
tff(fact_8901_ATP_Olambda__782,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_ij(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_782
tff(fact_8902_ATP_Olambda__783,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_sa(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_783
tff(fact_8903_ATP_Olambda__784,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A,Uuc: nat] : aa(nat,Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_uv(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub),Uuc) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uua,Uuc)),aa(nat,Aa,aa(Aa,fun(nat,Aa),power_power(Aa),aa(A,Aa,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_784
tff(fact_8904_ATP_Olambda__785,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_rx(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_785
tff(fact_8905_ATP_Olambda__786,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_gg(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_786
tff(fact_8906_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: set(A),Uuc: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aa(A,fun(set(A),fun(B,bool)),aTP_Lamp_xa(fun(B,A),fun(A,fun(set(A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uu,Uuc)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uub),aa(set(A),set(A),insert(A,Uua),bot_bot(set(A)))))) ) ) ).

% ATP.lambda_787
tff(fact_8907_ATP_Olambda__788,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_ahp(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uuc))
        & ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).

% ATP.lambda_788
tff(fact_8908_ATP_Olambda__789,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_adk(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_789
tff(fact_8909_ATP_Olambda__790,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_fp(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_790
tff(fact_8910_ATP_Olambda__791,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_fg(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_791
tff(fact_8911_ATP_Olambda__792,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_fj(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_792
tff(fact_8912_ATP_Olambda__793,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_abz(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_793
tff(fact_8913_ATP_Olambda__794,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_aac(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_794
tff(fact_8914_ATP_Olambda__795,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_qs(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),aa(real,real,cos(real),aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_795
tff(fact_8915_ATP_Olambda__796,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_rz(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_796
tff(fact_8916_ATP_Olambda__797,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_qy(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,exp(real),aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_797
tff(fact_8917_ATP_Olambda__798,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_ra(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,cos(real),aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_798
tff(fact_8918_ATP_Olambda__799,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_rn(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_799
tff(fact_8919_ATP_Olambda__800,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_qo(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_800
tff(fact_8920_ATP_Olambda__801,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_rj(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,uminus_uminus(real),aa(real,real,sin(real),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_801
tff(fact_8921_ATP_Olambda__802,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_qq(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_802
tff(fact_8922_ATP_Olambda__803,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_xi(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_803
tff(fact_8923_ATP_Olambda__804,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_xe(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))) ) ).

% ATP.lambda_804
tff(fact_8924_ATP_Olambda__805,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_wz(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uuc))),Uub))) ) ) ).

% ATP.lambda_805
tff(fact_8925_ATP_Olambda__806,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: filter(A),Uuc: A] : aa(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_xl(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_xk(A,A))))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_xk(A,A))))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_xk(A,A)))))) ) ).

% ATP.lambda_806
tff(fact_8926_ATP_Olambda__807,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_xg(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))) ) ).

% ATP.lambda_807
tff(fact_8927_ATP_Olambda__808,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_xf(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))) ) ).

% ATP.lambda_808
tff(fact_8928_ATP_Olambda__809,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_my(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),collect(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_mx(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_809
tff(fact_8929_ATP_Olambda__810,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_xj(fun(nat,A),fun(nat,fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),set_ord_atMost(nat,Uua))))) ) ) ).

% ATP.lambda_810
tff(fact_8930_ATP_Olambda__811,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(bool,fun(bool,A)),Uub: bool,Uuc: bool] : aa(bool,B,aa(bool,fun(bool,B),aa(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),aTP_Lamp_aie(fun(A,B),fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B))),Uu),Uua),Uub),Uuc) = aa(A,B,Uu,aa(bool,A,aa(bool,fun(bool,A),Uua,Uub),Uuc)) ).

% ATP.lambda_811
tff(fact_8931_ATP_Olambda__812,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_ib(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_812
tff(fact_8932_ATP_Olambda__813,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_hz(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_813
tff(fact_8933_ATP_Olambda__814,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_au(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_814
tff(fact_8934_ATP_Olambda__815,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_cn(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_815
tff(fact_8935_ATP_Olambda__816,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_aad(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_aac(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ).

% ATP.lambda_816
tff(fact_8936_ATP_Olambda__817,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_aca(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_abz(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ).

% ATP.lambda_817
tff(fact_8937_ATP_Olambda__818,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_kt(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_818
tff(fact_8938_ATP_Olambda__819,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_aem(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),product_Pair(list(A),product_prod(A,list(A)),takeWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),Uu),Uua)),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Uub),Uuc))) ).

% ATP.lambda_819
tff(fact_8939_ATP_Olambda__820,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: C,Uub: fun(C,A),Uuc: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(C,fun(fun(C,A),fun(C,A)),aTP_Lamp_rg(fun(C,A),fun(C,fun(fun(C,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua))),aa(C,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua)))) ) ).

% ATP.lambda_820
tff(fact_8940_ATP_Olambda__821,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_afy(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),product_Pair(list(A),product_prod(A,list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Uub),Uuc))) ).

% ATP.lambda_821
tff(fact_8941_ATP_Olambda__822,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_kv(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_822
tff(fact_8942_ATP_Olambda__823,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: fun(I6,fun(A,B)),Uuc: A,Uud: A] : aa(A,B,aa(A,fun(A,B),aa(fun(I6,fun(A,B)),fun(A,fun(A,B)),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B))),aTP_Lamp_rt(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(I6),B,aa(fun(I6,B),fun(set(I6),B),groups7311177749621191930dd_sum(I6,B),aa(A,fun(I6,B),aa(A,fun(A,fun(I6,B)),aa(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B)))),aTP_Lamp_rs(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_823
tff(fact_8943_ATP_Olambda__824,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_gd(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(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_gc(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_824
tff(fact_8944_ATP_Olambda__825,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_ql(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,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_qk(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_825
tff(fact_8945_ATP_Olambda__826,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_gh(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gg(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_826
tff(fact_8946_ATP_Olambda__827,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_gb(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_827
tff(fact_8947_ATP_Olambda__828,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_fw(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_828
tff(fact_8948_ATP_Olambda__829,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_fx(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_829
tff(fact_8949_ATP_Olambda__830,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_ahn(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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 ) )
          | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),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) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),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 )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),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_830
tff(fact_8950_ATP_Olambda__831,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_gc(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_831
tff(fact_8951_ATP_Olambda__832,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_rl(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_832
tff(fact_8952_ATP_Olambda__833,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_od(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uuc),Uud))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uud),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu)))
        & ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))
           => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I2)),X_13))
            <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Uub),I2)) ) )
        & ( ( Uuc = Uud )
         => ! [X4: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua)))
             => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
        & ( ( Uuc != Uud )
         => ( vEBT_V5917875025757280293ildren(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua,Uud)
            & ! [X4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),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(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua,X4)
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),X4))
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_833
tff(fact_8953_ATP_Olambda__834,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,A),Uua: C,Uub: fun(C,A),Uuc: int,Uud: C] : aa(C,A,aa(int,fun(C,A),aa(fun(C,A),fun(int,fun(C,A)),aa(C,fun(fun(C,A),fun(int,fun(C,A))),aTP_Lamp_abk(fun(C,A),fun(C,fun(fun(C,A),fun(int,fun(C,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,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(C,A,Uu,Uua),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uuc),one_one(int))))) ) ).

% ATP.lambda_834
tff(fact_8954_ATP_Olambda__835,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: fun(I6,fun(A,B)),Uuc: A,Uud: A,Uue: I6] : aa(I6,B,aa(A,fun(I6,B),aa(A,fun(A,fun(I6,B)),aa(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B)))),aTP_Lamp_rs(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,aa(I6,fun(A,B),Uub,Uue),Uud)),aa(set(I6),B,aa(fun(I6,B),fun(set(I6),B),groups7121269368397514597t_prod(I6,B),aa(A,fun(I6,B),aTP_Lamp_rq(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uua),Uuc)),aa(set(I6),set(I6),aa(set(I6),fun(set(I6),set(I6)),minus_minus(set(I6)),Uu),aa(set(I6),set(I6),insert(I6,Uue),bot_bot(set(I6)))))) ) ).

% ATP.lambda_835
tff(fact_8955_ATP_Olambda__836,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_re(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uua,Uue)),aa(C,A,Uuc,Uub))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uu,Uub)),aa(C,A,Uud,Uue)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uuc,Uub)),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_836
tff(fact_8956_ATP_Olambda__837,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_rv(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),aa(real,real,aa(real,fun(real,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_837
tff(fact_8957_ATP_Olambda__838,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_rp(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(C,A,Uu,Uub))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(C,A,Uuc,Uub))),aa(C,A,Uud,Uue))),aa(A,A,inverse_inverse(A),aa(C,A,Uuc,Uub))))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(C,A,Uua,Uue)),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_838
tff(fact_8958_ATP_Olambda__839,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT] : aa(vEBT_VEBT,B,aa(list(vEBT_VEBT),fun(vEBT_VEBT,B),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),aTP_Lamp_aid(fun(A,B),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,B,Uu,aa(vEBT_VEBT,A,aa(list(vEBT_VEBT),fun(vEBT_VEBT,A),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_839
tff(fact_8959_ATP_Olambda__840,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_afg(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_840
tff(fact_8960_ATP_Olambda__841,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_afl(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_841
tff(fact_8961_ATP_Olambda__842,axiom,
    ! [Uu: rat,Uua: nat] : aa(nat,rat,aTP_Lamp_os(rat,fun(nat,rat),Uu),Uua) = Uu ).

% ATP.lambda_842
tff(fact_8962_ATP_Olambda__843,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( zero(Aa)
        & topological_t2_space(Aa)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: Aa,Uua: A] : aa(A,Aa,aTP_Lamp_sh(Aa,fun(A,Aa),Uu),Uua) = Uu ) ).

% ATP.lambda_843
tff(fact_8963_ATP_Olambda__844,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_qu(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_844
tff(fact_8964_ATP_Olambda__845,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_acv(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_845
tff(fact_8965_ATP_Olambda__846,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aax(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_846
tff(fact_8966_ATP_Olambda__847,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bp(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_847
tff(fact_8967_ATP_Olambda__848,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ph(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_848
tff(fact_8968_ATP_Olambda__849,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_lc(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_849
tff(fact_8969_ATP_Olambda__850,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_acu(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_850
tff(fact_8970_ATP_Olambda__851,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_act(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_851
tff(fact_8971_ATP_Olambda__852,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_dh(complex,complex),Uu) = Uu ).

% ATP.lambda_852
tff(fact_8972_ATP_Olambda__853,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ai(nat,nat),Uu) = Uu ).

% ATP.lambda_853
tff(fact_8973_ATP_Olambda__854,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_bf(int,int),Uu) = Uu ).

% ATP.lambda_854
tff(fact_8974_ATP_Olambda__855,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [Uu: C] : aa(C,C,aTP_Lamp_ya(C,C),Uu) = Uu ) ).

% ATP.lambda_855
tff(fact_8975_ATP_Olambda__856,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_xk(A,A),Uu) = Uu ) ).

% ATP.lambda_856
tff(fact_8976_ATP_Olambda__857,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_pi(A,A),Uu) = Uu ) ).

% ATP.lambda_857
tff(fact_8977_ATP_Olambda__858,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_xq(A,A),Uu) = Uu ) ).

% ATP.lambda_858
tff(fact_8978_ATP_Olambda__859,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_xz(A,A),Uu) = Uu ) ).

% ATP.lambda_859
tff(fact_8979_ATP_Olambda__860,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_me(A,A),Uu) = Uu ) ).

% ATP.lambda_860
tff(fact_8980_ATP_Olambda__861,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ad(A,A),Uu) = Uu ) ).

% ATP.lambda_861
tff(fact_8981_ATP_Olambda__862,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ahk(A,A),Uu) = Uu ) ).

% ATP.lambda_862
tff(fact_8982_ATP_Olambda__863,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_np(A,A),Uu) = Uu ).

% ATP.lambda_863
tff(fact_8983_ATP_Olambda__864,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_afi(A,set(B)),Uu) = top_top(set(B)) ).

% ATP.lambda_864
tff(fact_8984_ATP_Olambda__865,axiom,
    ! [Uu: nat] : aa(nat,rat,aTP_Lamp_abq(nat,rat),Uu) = zero_zero(rat) ).

% ATP.lambda_865
tff(fact_8985_ATP_Olambda__866,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_bk(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_866
tff(fact_8986_ATP_Olambda__867,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_bc(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_867
tff(fact_8987_ATP_Olambda__868,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_cb(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_868
tff(fact_8988_ATP_Olambda__869,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_acs(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_869
tff(fact_8989_ATP_Olambda__870,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_qv(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_870
tff(fact_8990_ATP_Olambda__871,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_agi(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_871
tff(fact_8991_ATP_Olambda__872,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ah(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_872
tff(fact_8992_ATP_Olambda__873,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ahr(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_873
tff(fact_8993_ATP_Olambda__874,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ago(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_874
tff(fact_8994_ATP_Olambda__875,axiom,
    ! [Uu: nat] : aa(nat,rat,aTP_Lamp_abt(nat,rat),Uu) = one_one(rat) ).

% ATP.lambda_875
tff(fact_8995_ATP_Olambda__876,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_bd(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_876
tff(fact_8996_ATP_Olambda__877,axiom,
    ! [A: $tType,Uu: A] : aa(A,real,aTP_Lamp_lh(A,real),Uu) = one_one(real) ).

% ATP.lambda_877
tff(fact_8997_ATP_Olambda__878,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_ld(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_878
tff(fact_8998_ATP_Olambda__879,axiom,
    ! [B: $tType,D: $tType,Uu: B] : aa(B,option(D),aTP_Lamp_aeu(B,option(D)),Uu) = none(D) ).

% ATP.lambda_879
tff(fact_8999_ATP_Olambda__880,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_oj(B,option(A)),Uu) = none(A) ).

% ATP.lambda_880
tff(fact_9000_ATP_Olambda__881,axiom,
    ! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_nr(A,option(C)),Uu) = none(C) ).

% ATP.lambda_881
tff(fact_9001_ATP_Olambda__882,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_acf(A,option(B)),Uu) = none(B) ).

% ATP.lambda_882
tff(fact_9002_ATP_Olambda__883,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_hl(real,bool),Uu))
    <=> $false ) ).

% ATP.lambda_883
tff(fact_9003_ATP_Olambda__884,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ag(nat,bool),Uu))
    <=> $false ) ).

% ATP.lambda_884
tff(fact_9004_ATP_Olambda__885,axiom,
    ! [B: $tType,Uu: B] :
      ( pp(aa(B,bool,aTP_Lamp_agy(B,bool),Uu))
    <=> $false ) ).

% ATP.lambda_885
tff(fact_9005_ATP_Olambda__886,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_nf(A,bool),Uu))
    <=> $false ) ).

% ATP.lambda_886
tff(fact_9006_ATP_Olambda__887,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_af(nat,bool),Uu))
    <=> $true ) ).

% ATP.lambda_887
tff(fact_9007_ATP_Olambda__888,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_ne(A,bool),Uu))
    <=> $true ) ).

% ATP.lambda_888
tff(fact_9008_ATP_Olambda__889,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(nat,nat),aTP_Lamp_afm(B,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_889
tff(fact_9009_ATP_Olambda__890,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_aea(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_890

% Type constructors (884)
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_1,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_2,axiom,
    ! [A11: $tType] : bounded_lattice(filter(A11)) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_3,axiom,
    bounded_lattice(bool) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_4,axiom,
    ! [A11: $tType] : bounded_lattice(set(A11)) ).

tff(tcon_fun___Lattices_Obounded__lattice_5,axiom,
    ! [A11: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounded_lattice(fun(A11,A15)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A11: $tType,A15: $tType] :
      ( comple6319245703460814977attice(A15)
     => condit1219197933456340205attice(fun(A11,A15)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A11: $tType,A15: $tType] :
      ( counta3822494911875563373attice(A15)
     => counta3822494911875563373attice(fun(A11,A15)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A11: $tType,A15: $tType] :
      ( comple592849572758109894attice(A15)
     => comple592849572758109894attice(fun(A11,A15)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A11: $tType,A15: $tType] :
      ( comple489889107523837845lgebra(A15)
     => comple489889107523837845lgebra(fun(A11,A15)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A11: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounde4967611905675639751up_bot(fun(A11,A15)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A11: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounde4346867609351753570nf_top(fun(A11,A15)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A11: $tType,A15: $tType] :
      ( comple6319245703460814977attice(A15)
     => comple6319245703460814977attice(fun(A11,A15)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A11: $tType,A15: $tType] :
      ( boolea8198339166811842893lgebra(A15)
     => boolea8198339166811842893lgebra(fun(A11,A15)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A11: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounded_lattice_top(fun(A11,A15)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A11: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounded_lattice_bot(fun(A11,A15)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A11: $tType,A15: $tType] :
      ( semilattice_sup(A15)
     => semilattice_sup(fun(A11,A15)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A11: $tType,A15: $tType] :
      ( semilattice_inf(A15)
     => semilattice_inf(fun(A11,A15)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A11: $tType,A15: $tType] :
      ( distrib_lattice(A15)
     => distrib_lattice(fun(A11,A15)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A11: $tType,A15: $tType] :
      ( order_top(A15)
     => order_top(fun(A11,A15)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A11: $tType,A15: $tType] :
      ( order_bot(A15)
     => order_bot(fun(A11,A15)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A11: $tType,A15: $tType] :
      ( preorder(A15)
     => preorder(fun(A11,A15)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A11: $tType,A15: $tType] :
      ( lattice(A15)
     => lattice(fun(A11,A15)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A11: $tType,A15: $tType] :
      ( order(A15)
     => order(fun(A11,A15)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A11: $tType,A15: $tType] :
      ( ord(A15)
     => ord(fun(A11,A15)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A11: $tType,A15: $tType] :
      ( uminus(A15)
     => uminus(fun(A11,A15)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A11: $tType,A15: $tType] :
      ( minus(A15)
     => minus(fun(A11,A15)) ) ).

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___Rings_Onormalization__semidom__multiplicative,axiom,
    normal6328177297339901930cative(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___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_Osemidom__divide__unit__factor,axiom,
    semido2269285787275462019factor(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___Lattices_Odistrib__lattice_9,axiom,
    distrib_lattice(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
    semiring_1_cancel(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_10,axiom,
    preorder(int) ).

tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
    linorder(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult(int) ).

tff(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
    idom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Oidom__divide,axiom,
    idom_divide(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top(int) ).

tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot(int) ).

tff(tcon_Int_Oint___Lattices_Olattice_11,axiom,
    lattice(int) ).

tff(tcon_Int_Oint___Groups_Ogroup__add,axiom,
    group_add(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
    semiring_gcd(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
    semiring_Gcd(int) ).

tff(tcon_Int_Oint___Rings_Omult__zero,axiom,
    mult_zero(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring(int) ).

tff(tcon_Int_Oint___Orderings_Oorder_12,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Rings_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_13,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_14,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Groups_Ominus_15,axiom,
    minus(int) ).

tff(tcon_Int_Oint___GCD_Oring__gcd,axiom,
    ring_gcd(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_16,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_17,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_18,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_19,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_20,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_21,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_22,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_23,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_24,axiom,
    normal6328177297339901930cative(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_25,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_26,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_27,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_28,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_29,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_30,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_31,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_32,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_33,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_34,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_35,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_36,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_37,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_38,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_39,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_40,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_41,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_42,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_43,axiom,
    semido2269285787275462019factor(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_44,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_45,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_46,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_47,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_48,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_49,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_50,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_51,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_52,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_53,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_54,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_55,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_56,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom_57,axiom,
    normal8620421768224518004emidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_58,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_59,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_60,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_61,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_62,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_63,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_64,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_65,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_66,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_67,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_68,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_69,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_70,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_71,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_72,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_73,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_74,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_75,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_76,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_77,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_78,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_79,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_80,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_81,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_82,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_83,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_84,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_85,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_86,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_87,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_88,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_89,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_90,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_91,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_92,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_93,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_94,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_95,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_96,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_97,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_98,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_99,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_100,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_101,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_102,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_103,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_104,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_105,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_106,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_107,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_108,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_109,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_110,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_111,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_112,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_113,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_114,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_115,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_116,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_117,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_118,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_119,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_120,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_121,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_122,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_123,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_124,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_125,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_126,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_127,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_128,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_129,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_130,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_131,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_132,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_133,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_134,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_135,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_136,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_137,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_138,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_139,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_140,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_141,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_142,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_143,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_144,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_145,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_146,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_147,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_148,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_149,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_150,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_151,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_152,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_153,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_154,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_155,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_156,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_157,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_158,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_159,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_160,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_161,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_162,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_163,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_164,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_165,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_166,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_167,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_168,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_169,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_170,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_171,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_172,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_173,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_174,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_175,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_176,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_177,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_178,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_179,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_180,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_181,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_182,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_183,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_184,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_185,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_186,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_187,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_188,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_189,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_190,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_191,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_192,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_193,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_194,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_195,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_196,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_197,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_198,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_199,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_200,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_201,axiom,
    ! [A11: $tType] : condit1219197933456340205attice(set(A11)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_202,axiom,
    ! [A11: $tType] : counta3822494911875563373attice(set(A11)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_203,axiom,
    ! [A11: $tType] : comple592849572758109894attice(set(A11)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_204,axiom,
    ! [A11: $tType] : comple489889107523837845lgebra(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_205,axiom,
    ! [A11: $tType] : bounde4967611905675639751up_bot(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_206,axiom,
    ! [A11: $tType] : bounde4346867609351753570nf_top(set(A11)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_207,axiom,
    ! [A11: $tType] : comple6319245703460814977attice(set(A11)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_208,axiom,
    ! [A11: $tType] : boolea8198339166811842893lgebra(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_209,axiom,
    ! [A11: $tType] : bounded_lattice_top(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_210,axiom,
    ! [A11: $tType] : bounded_lattice_bot(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_211,axiom,
    ! [A11: $tType] : semilattice_sup(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_212,axiom,
    ! [A11: $tType] : semilattice_inf(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_213,axiom,
    ! [A11: $tType] : distrib_lattice(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_214,axiom,
    ! [A11: $tType] : order_top(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_215,axiom,
    ! [A11: $tType] : order_bot(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_216,axiom,
    ! [A11: $tType] : preorder(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Olattice_217,axiom,
    ! [A11: $tType] : lattice(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder_218,axiom,
    ! [A11: $tType] : order(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oord_219,axiom,
    ! [A11: $tType] : ord(set(A11)) ).

tff(tcon_Set_Oset___Groups_Ouminus_220,axiom,
    ! [A11: $tType] : uminus(set(A11)) ).

tff(tcon_Set_Oset___Groups_Ominus_221,axiom,
    ! [A11: $tType] : minus(set(A11)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_222,axiom,
    condit1219197933456340205attice(bool) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_223,axiom,
    counta3822494911875563373attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_224,axiom,
    comple592849572758109894attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_225,axiom,
    comple489889107523837845lgebra(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_226,axiom,
    topolo4958980785337419405_space(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_227,axiom,
    topolo1944317154257567458pology(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_228,axiom,
    bounde4967611905675639751up_bot(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_229,axiom,
    bounde4346867609351753570nf_top(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_230,axiom,
    comple6319245703460814977attice(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_231,axiom,
    topolo2564578578187576103pology(bool) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_232,axiom,
    boolea8198339166811842893lgebra(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_233,axiom,
    bounded_lattice_top(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_234,axiom,
    bounded_lattice_bot(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_235,axiom,
    topological_t2_space(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_236,axiom,
    semilattice_sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_237,axiom,
    semilattice_inf(bool) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_238,axiom,
    distrib_lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_239,axiom,
    order_top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_240,axiom,
    order_bot(bool) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_241,axiom,
    preorder(bool) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_242,axiom,
    linorder(bool) ).

tff(tcon_HOL_Obool___Lattices_Olattice_243,axiom,
    lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder_244,axiom,
    order(bool) ).

tff(tcon_HOL_Obool___Orderings_Oord_245,axiom,
    ord(bool) ).

tff(tcon_HOL_Obool___Groups_Ouminus_246,axiom,
    uminus(bool) ).

tff(tcon_HOL_Obool___Groups_Ominus_247,axiom,
    minus(bool) ).

tff(tcon_List_Olist___Nat_Osize_248,axiom,
    ! [A11: $tType] : size(list(A11)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_249,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_250,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_251,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_252,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_253,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_254,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_255,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_256,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_257,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_258,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_259,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_260,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_261,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_262,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_263,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_264,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_265,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_266,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_267,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_268,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_269,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_270,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_271,axiom,
    semiri3467727345109120633visors(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_272,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_273,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_274,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_275,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_276,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_277,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_278,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_279,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_280,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_281,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_282,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Odist__norm,axiom,
    real_V6936659425649961206t_norm(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_283,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_284,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_285,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_286,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_287,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_288,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_289,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_290,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_291,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_292,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_293,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_294,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_295,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_296,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_297,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_298,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_299,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_300,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_301,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_302,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_303,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_304,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_305,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_306,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_307,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_308,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_309,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_310,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_311,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_312,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_313,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_314,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_315,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_316,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_317,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_318,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_319,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_320,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_321,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_322,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_323,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_324,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_325,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_326,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_327,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_328,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_329,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_330,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_331,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_332,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_333,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_334,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_335,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_336,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_337,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_338,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_339,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_340,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_341,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_342,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_343,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_344,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_345,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_346,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_347,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_348,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_349,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_350,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_351,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_352,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_353,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_354,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_355,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Nat_Osize_356,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_357,axiom,
    ! [A11: $tType,A15: $tType] : size(sum_sum(A11,A15)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_358,axiom,
    ! [A11: $tType] : condit1219197933456340205attice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_359,axiom,
    ! [A11: $tType] : counta3822494911875563373attice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_360,axiom,
    ! [A11: $tType] : bounde4967611905675639751up_bot(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_361,axiom,
    ! [A11: $tType] : bounde4346867609351753570nf_top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_362,axiom,
    ! [A11: $tType] : comple6319245703460814977attice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_363,axiom,
    ! [A11: $tType] : bounded_lattice_top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_364,axiom,
    ! [A11: $tType] : bounded_lattice_bot(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_365,axiom,
    ! [A11: $tType] : semilattice_sup(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_366,axiom,
    ! [A11: $tType] : semilattice_inf(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_367,axiom,
    ! [A11: $tType] : distrib_lattice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_368,axiom,
    ! [A11: $tType] : order_top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_369,axiom,
    ! [A11: $tType] : order_bot(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_370,axiom,
    ! [A11: $tType] : preorder(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_371,axiom,
    ! [A11: $tType] : lattice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_372,axiom,
    ! [A11: $tType] : order(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_373,axiom,
    ! [A11: $tType] : ord(filter(A11)) ).

tff(tcon_Option_Ooption___Nat_Osize_374,axiom,
    ! [A11: $tType] : size(option(A11)) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_375,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_376,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_377,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_378,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_379,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_380,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_381,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___Groups_Oplus_382,axiom,
    plus(literal) ).

tff(tcon_String_Oliteral___Nat_Osize_383,axiom,
    size(literal) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_384,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_385,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_386,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_387,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_388,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_389,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_390,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_391,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_392,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_393,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_394,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_395,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_396,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_397,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_398,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_399,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_400,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_401,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_402,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_403,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_404,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_405,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_406,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_407,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_408,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_409,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Odist__norm_410,axiom,
    real_V6936659425649961206t_norm(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_411,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_412,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_413,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_414,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_415,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_416,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_417,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_418,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_419,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_420,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_421,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_422,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_423,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_424,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_425,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_426,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_427,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_428,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_429,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_430,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_431,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_432,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_433,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_434,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_435,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_436,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_437,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_438,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_439,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_440,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_441,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_442,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_443,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_444,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_445,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_446,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_447,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_448,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_449,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_450,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_451,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_452,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_453,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_454,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_455,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_456,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_457,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_458,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_459,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_460,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_461,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_462,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_463,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_464,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_465,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_466,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_467,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_468,axiom,
    bounde4346867609351753570nf_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_469,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_470,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_471,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_472,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_473,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_474,axiom,
    bounded_lattice_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_475,axiom,
    bounded_lattice_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_476,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_477,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_478,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_479,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_480,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_481,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_482,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_483,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_484,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_485,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_486,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_487,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_488,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_489,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_490,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_491,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_492,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_493,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_494,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_495,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_496,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_497,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_498,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_499,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_500,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_501,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_502,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_503,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_504,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_505,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_506,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_507,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_508,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_509,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_510,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_511,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_512,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_513,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_514,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_515,axiom,
    ! [A11: $tType,A15: $tType] :
      ( ( topolo4958980785337419405_space(A11)
        & topolo4958980785337419405_space(A15) )
     => topolo4958980785337419405_space(product_prod(A11,A15)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_516,axiom,
    ! [A11: $tType,A15: $tType] :
      ( ( topological_t2_space(A11)
        & topological_t2_space(A15) )
     => topological_t2_space(product_prod(A11,A15)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_517,axiom,
    ! [A11: $tType,A15: $tType] : size(product_prod(A11,A15)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_518,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_519,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_520,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_521,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_522,axiom,
    comple489889107523837845lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_523,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_524,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_525,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_526,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_527,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_528,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_529,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_530,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_531,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_532,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_533,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_534,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_535,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_536,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_537,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_538,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_539,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_540,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_541,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_542,axiom,
    minus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_543,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_544,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_545,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_546,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_547,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_548,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_549,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_550,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_551,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_552,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_553,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_554,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_555,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_556,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_557,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_558,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_559,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_560,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_561,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_562,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_563,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_564,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_565,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_566,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_567,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_568,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_569,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_570,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_571,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_572,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_573,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_574,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_575,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_576,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_577,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_578,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_579,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_580,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_581,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_582,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_583,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_584,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_585,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_586,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_587,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_588,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_589,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_590,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_591,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_592,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_593,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_594,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_595,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_596,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_597,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_598,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_599,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_600,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_601,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_602,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_603,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_604,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_605,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_606,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_607,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_608,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_609,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_610,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_611,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_612,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_613,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_614,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_615,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_616,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_617,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_618,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_619,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_620,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_621,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_622,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_623,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_624,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_625,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_626,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_627,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_628,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_629,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_630,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_631,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_632,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_633,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_634,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_635,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_636,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_637,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_638,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_639,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_640,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_641,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_642,axiom,
    bit_un5681908812861735899ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_643,axiom,
    euclid5411537665997757685th_nat(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_644,axiom,
    ordere1937475149494474687imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_645,axiom,
    euclid3128863361964157862miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_646,axiom,
    euclid4440199948858584721cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_647,axiom,
    semiri6575147826004484403cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_648,axiom,
    strict9044650504122735259up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_649,axiom,
    ordere580206878836729694up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_650,axiom,
    ordere2412721322843649153imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_651,axiom,
    bit_se359711467146920520ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_652,axiom,
    linord2810124833399127020strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_653,axiom,
    strict7427464778891057005id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_654,axiom,
    ordere8940638589300402666id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_655,axiom,
    euclid3725896446679973847miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_656,axiom,
    semiri2026040879449505780visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_657,axiom,
    linord181362715937106298miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_658,axiom,
    linord8928482502909563296strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_659,axiom,
    semiri3467727345109120633visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_660,axiom,
    ordere6658533253407199908up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_661,axiom,
    ordere6911136660526730532id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_662,axiom,
    cancel2418104881723323429up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_663,axiom,
    cancel1802427076303600483id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_664,axiom,
    comm_s4317794764714335236cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_665,axiom,
    bit_semiring_bits(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_666,axiom,
    ordere2520102378445227354miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_667,axiom,
    cancel_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_668,axiom,
    linordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_669,axiom,
    ordered_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_670,axiom,
    linordered_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_671,axiom,
    ab_semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_672,axiom,
    semiring_1_cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_673,axiom,
    algebraic_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_674,axiom,
    comm_monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_675,axiom,
    comm_monoid_diff(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_676,axiom,
    ab_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_677,axiom,
    ordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_678,axiom,
    semiring_parity(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_679,axiom,
    comm_monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_680,axiom,
    semiring_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_681,axiom,
    comm_semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_682,axiom,
    comm_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_683,axiom,
    semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_684,axiom,
    semidom_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_685,axiom,
    semidom_divide(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_686,axiom,
    semiring_numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_687,axiom,
    semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_688,axiom,
    zero_less_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_689,axiom,
    comm_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_690,axiom,
    semiring_char_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_691,axiom,
    zero_neq_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_692,axiom,
    preorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_693,axiom,
    linorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_694,axiom,
    monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_695,axiom,
    monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_696,axiom,
    semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_697,axiom,
    semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_698,axiom,
    mult_zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_699,axiom,
    order(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_700,axiom,
    semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom_701,axiom,
    semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oord_702,axiom,
    ord(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ominus_703,axiom,
    minus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Power_Opower_704,axiom,
    power(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Onumeral_705,axiom,
    numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ozero_706,axiom,
    zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oplus_707,axiom,
    plus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oone_708,axiom,
    one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Odvd_709,axiom,
    dvd(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osize_710,axiom,
    size(code_natural) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_711,axiom,
    size(vEBT_VEBT) ).

% Helper facts (24)
tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y2: A] : if(A,fFalse,X,Y2) = Y2 ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y2: A] : if(A,fTrue,X,Y2) = X ).

tff(help_fEx_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(aa(A,bool,P,X))
      | pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(fAll(A,P))
      | pp(aa(A,bool,P,X)) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa(bool,bool,fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa(bool,bool,fNot,P))
      | ~ pp(P) ) ).

tff(help_COMBB_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(B,C),Q: fun(A,B),R2: A] : aa(A,C,combb(B,C,A,P,Q),R2) = aa(B,C,P,aa(A,B,Q,R2)) ).

tff(help_COMBC_1_1_U,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,fun(B,C)),Q: B,R2: A] : aa(A,C,combc(A,B,C,P,Q),R2) = aa(B,C,aa(A,fun(B,C),P,R2),Q) ).

tff(help_COMBS_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,fun(B,C)),Q: fun(A,B),R2: A] : aa(A,C,combs(A,B,C,P,Q),R2) = aa(B,C,aa(A,fun(B,C),P,R2),aa(A,B,Q,R2)) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(fconj(P,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fdisj(P,Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(fdisj(P,Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(fdisj(P,Q)) ) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y2: A] :
      ( ( X != Y2 )
      | pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y2)) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y2: A] :
      ( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y2))
      | ( X = Y2 ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(A,bool,P,fChoice(A,P)) = aa(fun(A,bool),bool,fEx(A),P) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    vEBT_invar_vebt(t,n) ).

tff(conj_1,conjecture,
    ( pp(aa(nat,bool,vEBT_vebt_member(t),x))
  <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),x),vEBT_set_vebt(t))) ) ).

%------------------------------------------------------------------------------